The first six books of the Elements of Euclid : in which coloured diagrams and symbols are used instead of letters for the greater ease of learners

The Dihner Library
of the History of
Science mid Technology

SMITHSONIAN INSTITUTION LIBRARIES

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BYRNE’S EUCLID

THE FIRST SIX BOOKS OF

THE ELEMENTS OF EUCLID

WITH COLOURED DIAGRAMS

AND SYMBOLS

THE FIRST SIX BOOKS OF

THE ELEMENTS OF EUCLID

IN WHICH COLOURED DIAGRAMS AND SYMBOLS
ARE USED INSTEAD OF LETTERS FOR THE
GREATER EASE OF LEARNERS

BY OLIVER BYRNE

SURVEYOR OF HER MAJESTY’S SETTLEMENTS IN THE FALKLAND ISLANDS
AND AUTHOR OF NUMEROUS MATHEMATICAL WORKS

LONDON

WILLIAM PICKERING

1847

TO THE

RIGHT HONOURABLE THE EARL FITZWILLIAM,

ETC. ETC. ETC.

THIS WORK IS DEDICATED

BY HIS LORDSHIP'S OBEDIENT

AND MUCH OBLIGED SERVANT,

OLIVER BYRNE.

INTRODUCTION.

HE arts and faiences have become fo extenlive,
that to facilitate their acquirement is of as
much importance as to extend their boundaries.
Illustration, if it does not fhorten the time of
Study, will at leaSt make it more agreeable. This Work
has a greater aim than mere illustration; we do not intro¬
duce colours for the purpofe of entertainment, or to amufe
by certain combinations of tint and form , but to aifiSt the
mind in its refearches after truth, to increafe the facilities
of instruction, and to diffufe permanent knowledge. If we
wanted authorities to prove the importance and ufefulnefs
of geometry, we might quote every philofopher Since the
days of Plato. Among the Greeks, in ancient, as in the
fchool of Pettalozzi and others in recent times, geometry
was adopted as the beft gymnaStic of the mind. In fail,
Euclid’s Elements have become, by common confent, the
bafis of mathematical fcience all over the civilized globe.
But this will not appear extraordinary, if we consider that
this fublime fcience is not only better calculated than any
other to call forth the fpirit of inquiry, to elevate the mind,
and to Strengthen the reafoning faculties, but alfo it forms
the beft introduction to molt of the ufeful and important
vocations of human life. Arithmetic, land-furveying, men-
furation, engineering, navigation, mechanics, hydrostatics,
pneumatics, optics, physical aStronomy, &c. are all depen¬
dent on the proportions of geometry.

INTRODUCTION .

vm

Much however depends on the firft communication of
any fcience to a learner, though the beft and moft eafy
methods are feldom adopted. Proportions are placed be¬
fore a ftudent, who though having a fufficient underftand-
ing, is told juft as much about them on entering at the
very threfhold of the fcience, as gives him a prepofleflion
moft unfavourable to his future ftudy of this delightful
fubjedt; or “ the formalities and paraphernalia of rigour are
fo oftentatioufty put forward, as almoft to hide the reality.
Endlefs and perplexing repetitions, which do not confer
greater exactitude on the reafoning, render the demonftra-
tions involved and obfcure, and conceal from the view of
the ftudent the confecution of evidence.” Thus an aver-
lion is created in the mind of the pupil, and a fubjedt fo
calculated to improve the reafoning powers, and give the
habit of clofe thinking, is degraded by a dry and rigid
courfe of inftrudtion into an uninterefting exercife of the
memory. To raife the curiolity, and to awaken the liftlefs
and dormant powers of younger minds fhould be the aim
of every teacher; but where examples of excellence are
wanting, the attempts to attain it are but few, while emi¬
nence excites attention and produces imitation. The objedt
of this Work is to introduce a method of teaching geome¬
try, which has been much approved of by many fcientific
men in this country, as well as in France and America.
The plan here adopted forcibly appeals to the eye, the moft
fenfitive and the moft comprehenftve of our external organs,
and its pre-eminence to imprint it fubjedt on the mind is
fupported by the incontrovertible maxim exprefted in the
well known words of Horace :—

Segnius irritant ani?nos demijfa per aure7n
Ipuam qua funt oculis fubjedta fidelibus.

A feebler imprefs through the ear is made,

Than what is by the faithful eye conveyed.

DSl

INTRODUCTION.

IX

All language conlifts of reprefentative ligns, and thofe
figns are the belt which effect their purpofes with the
greated precilion and dilpatch. Such for all common pur¬
pofes are the audible ligns called words, which are drill
conlidered as audible, whether addrelfed immediately to the
ear, or through the medium of letters to the eye. Geo¬
metrical diagrams are not ligns, but the materials of geo¬
metrical fcience, the objed of which is to Ihow the relative
quantities of their parts by a procefs of reafoning called
Demonftration. This reafoning has been generally carried
on by words, letters, and black or uncoloured diagrams;
but as the ufe of coloured lymbols, ligns, and diagrams in
the linear arts and fciences, renders the procefs of reafon¬
ing more precife, and the attainment more expeditious, they
have been in this inflance accordingly adopted.

Such is the expedition of this enticing mode of commu¬
nicating knowledge, that the Elements of Euclid can be
acquired in lefs than one third the time ufually employed,
and the retention by the memory is much more permanent;
thefe facts have been afcertained by numerous experiments
made by the inventor, and feveral others who have adopted
his pi ans. The particulars of which are few and obvious;
the letters annexed to points, lines, or other parts of a dia¬
gram are in fad: but arbitrary names, and reprefent them in
the demonftration ; inftead of thefe, the parts being differ¬
ently coloured, are made ^

to name themfelves, for

their forms in correfpond- \

ing colours represent them ^

in the demonftration.

In order to give a bet-

ter idea of this fyftem, and A {

of the advantages gained by its adoption, let us take a right

X

INTRODUCTION.

angled triangle, and exprefs fame of its properties both by
colours and the method generally employed.

Some of the properties of the right angled triangle ABC,
expreffed by the method generally employed.

1. The angle BAG, together with the angles BCA and
ABC are equal to two right angles, or twice the angle ABC.

2. The angle CAB added to the angle ACB will be equal
to the angle ABC.

3. The angle ABC is greater than either of the angles
BAC or BCA.

4. The angle BCA or the angle CAB is lefs than the
angle ABC.

5. If from the angle ABC, there be taken the angle
BAC, the remainder will be equal to the angle ACB.

6. The fquare of AC is equal to the fum of the fquares
of AB and BC.

The fame properties expreffed by colouring the different parts.

a + = 2 = fT\ .

That is, the red angle added to the yellow angle added to
the blue angle, equal twice the yellow angle, equal two
right angles.

Or in words, the red angle added to the blue angle, equal
the yellow angle.

The yellow angle is greater than either the red or blue
angle.

INTRODUCTION.

xi

Either the red or blue angle is lefs than the yellow angle.

In other terms, the yellow angle made lefs by the blue angle
equal the red angle.

6 .

+

That is, the fquare of the yellow line is equal to the fum
of the fquares of the blue and red lines.

In oral demonftrations we gain with colours this impor¬
tant advantage, the eye and the ear can be addreffed at the
fame moment, fo that for teaching geometry, and other
linear arts and fciences, in claifes, the fyftem is the beft ever
propofed, this is apparent from the examples juft given.

Whence it is evident that a reference from the text to
the diagram is more rapid and fure, by giving the forms
and colours of the parts, or by naming the parts and their
colours, than naming the parts and letters on the diagram.
Befides the fuperior iimplicity, this lyftem is like wife con-
fpicuous for concentration, and wholly excludes the injuri¬
ous though prevalent practice of allowing the ftudent to
commit the demonflration to memory; until reafon, and fadt,
and proof only make impreffions on the underftanding.

Again, when lecturing on the principles or properties of
figures, if we mention the colour of the part or parts re¬
ferred to, as in faying, the red angle, the blue line, or lines,
&c. the part or parts thus named will be immediately feen
by all in the clafs at the fame inftant; not fo if we fay the
angle ABC, the triangle PFQ^the figure EGKt, and fo on;

Xll

INTRODUCTION.

for the letters mufl be traced one by one before the ffudents
arrange in their minds the particular magnitude referred to,
which often occafions confulion and error, as well as lofs of
time. Alfo if the parts which are given as equal, have the
fame colours in any diagram, the mind will not wander
from the objedt before it; that is, fuch an arrangement pre-
fents an ocular demonffration of the parts to be proved
equal, and the learner retains the data throughout the whole
of the reafoning. But whatever may be the advantages of
the prefent plan, if it be not fubftituted for, it can always
be made a powerful auxiliary to the other methods, for the
purpofe of introduction, or of a more fpeedy reminifcence,
or of more permanent retention by the memory.

The experience of all who have formed lyffems to im-
prefs fads on the underftanding, agree in proving that
coloured reprefentations, as pidures, cuts, diagrams, &c. are
more eaiily fixed in the mind than mere fentences un¬
marked by any peculiarity. Curious as it may appear,
poets feem to be aware of this fad more than mathema¬
ticians ; many modern poets allude to this vifible fyftem of
communicating knowledge, one of them has thus exprefied
himfelf:

Sounds which addrefs the ear are loft and die
In one fhort hour, but thefe which ftrike the eye,

Live long upon the mind, the faithful fight
Engraves the knowledge with a beam of light.

This perhaps may be reckoned the only improvement
which plain geometry has received fince the days of Euclid,
and if there were any geometers of note before that time,
Euclid’s fuccefs has quite eclipfed their memory, and even
occafioned all good things of that kind to be afiigned to
him; like JEfop among the writers of Fables. It may
alfo be worthy of remark, as tangible diagrams afford the
only medium through which geometry and other linear

INTRODUCTION.

• • •
Xlll

arts and fciences can be taught to the blind, this vifible fys-
tem is no lefs adapted to the exigencies of the deaf and
dumb.

Care mult be taken to fhow that colour has nothing to
do with the lines, angles, or magnitudes, except merely to
name them. A mathematical line, which is length with¬
out breadth, cannot polfefs colour, yet the junction of two
colours on the fame plane gives a good idea of what is
meant by a mathematical line; recoiled: we are fpeaking
familiarly, fuch a junction is to be underftood and not the
colour, when we fay the black line, the red line or lines, &c.

Colours and coloured diagrams may at firft appear a
clumfy method to convey proper notions of the properties
and parts of mathematical figures and magnitudes, how¬
ever they will be found to afford a means more refined and
extenfive than any that has been hitherto propofed.

We fhall here define a point, a line, and a furface, and
demonftrate apropofition in order to fhow the truth of this
affertion.

A point is that which has pofition, but not magnitude;
or a point is pofition only, abftradted from the confideration
of length, breadth, and thicknefs. Perhaps the follow¬
ing defcription is better calculated to explain the nature of
a mathematical point to thofe who have not acquired the
idea, than the above fpecious definition.

Let three colours meet and cover a
portion of the paper, where they meet
is not blue, nor is it yellow, nor is it
red, as it occupies no portion of the
plane, for if it did, it would belong
to the blue, the red, or the yellow
pait; yet it exifts, and has pofition
without magnitude, fo that with a little reflection, this June-

XIV

INTRODUCTION.

tion of three colours on a plane, gives a good idea of a
mathematical point.

A line is length without breadth. With the afliftance
of colours, nearly in the fame manner as before, an idea of
a line may be thus given:—

Let two colours meet and cover a portion of the paper;

where they meet is not red, nor is it
blue; therefore the junction occu¬
pies no portion of the plane, and
therefore it cannot have breadth, but
only length: from which we can
readily form an idea of what is meant by a mathematical
line. For the purpofe of illuftration, one colour differing
from the colour of the paper, or plane upon which it is
drawn, would have been fufficient; hence in future, if we
fay the red line, the blue line, or lines, &c. it is the junc¬
tions with the plane upon which they are drawn are to be
underflood.

Surface is that which has length and breadth without
thicknefs.

without thicknefs, and

When we confider a folid body
(PQ), we perceive at once that it
has three dimenfions, namely :—
length, breadth, and thicknefs;
fuppofeone part of this folid (PS)
to be red, and the other part (QR)
yellow, and that the colours be
diftindt without commingling, the
blue furface (RS) which feparates
thefe parts, or which is the fame
thing, that which divides the folid
without lofs of material, mu ft be
only poffeffes length and breadth;

INTRODUCTION.

xv

this plainly appears from reafoning, iimilar to that juft em¬
ployed in defining, or rather defcribing a point and a line.

The propofition which we have feledted to elucidate the
manner in which the principles are applied, is the fifth of
the firft Book.

In an ifofceles triangle ABC, the
internal angles at the bafe ABC,
ACB are equal, and when the fides
AB, AC are produced, the exter¬
nal angles at the bafe BCE, CBD
are alfo equal.

Produce . ■— and —
make zz

(B. i.pr. 3.)

in

and

we have

• •

and

and

>= ^

.4^^ common :

(B. 1. pr. 4.)

Again in

7

and

E

9

XVJ

INTRODUCTION .

and

• •

and

:= (B. i. pr. 4).

E. D.

By annexing Letters to the Diagram.

Let the equal lides AB and AC be produced through the
extremities BC, of the third fide, and in the produced part
BD of either, let any point D be affumed, and from the
other let AE be cut off equal to AD (B. 1. pr. 3). Let
the points E and D, fo taken in the produced fides, be con¬
nected by ftraight lines DC and BE with the alternate ex¬
tremities of the third lide of the triangle.

In the triangles DAC and EAB the tides DA and AC
are refpeCtively equal to EA and AB, and the included
angle A is common to both triangles. Hence (B. 1. pr. 4.)
the line DC is equal to BE, the angle ADC to the angle
AEB, and the angle ACD to the angle ABE; if from
the equal lines AD and AE the equal fides AB and AC
be taken, the remainders BD and CE will be equal. Hence
in the triangles BDC and CEB, the fides BD and DC are
refpeCtively equal to CE and EB, and the angles D and E
included by thofe fides are alfo equal. Hence (B. 1. pr. 4.)

INTRODUCTION .

XVII

the angles DEC and ECB, which are thofe included by
the third fide BC and the productions of the equal fides
AB and AC are equal. Alfo the angles DCB and EBC
are equal if thofe equals be taken from the angles DCA
and EBA before proved equal, the remainders, which are
the angles ABC and ACB oppofite to the equal lides, will
be equal.

Therefore in an ifofceles triangle , &c.

Q^JE. D.

Our object in this place being to introduce the fyftem
rather than to teach any particular fet of proportions, we
have therefore feledted the foregoing out of the regular
courfe. For fchools and other public places of inftrudlion,
dyed chalks will anfwer to defcribe diagrams, &c. for private
ufe coloured pencils will be found very convenient.

We are happy to find that the Elements of Mathematics
now forms a confiderable part of every found female edu¬
cation, therefore we call the attention of thofe interefted
or engaged in the education of ladies to this very attractive
mode of communicating knowledge, and to the fucceeding
work for its future developement.

We fhall for the prefent conclude by obferving, as the
fenfes of fight and hearing can be fo forcibly and inftanta-
neously addrefied alike with one thoufand as with one, the
million might be taught geometry and other branches of
mathematics with great eafe, this would advance the pur-
pofe of education more than any thing that might be named,
for it would teach the people how to think, and not what
to think ; it is in this particular the great error of education
originates.

w

d

XV111

THE ELEMENTS OF EUCLID.

BOOK I.

DEFINITIONS.

I.

A point is that which has no parts.

II.

A line is length without breadth.

III.

The extremities of a line are points.

IV.

A ftraight or right line is that which lies evenly between

its extremities.

V.

A furface is that which has length and breadth only.

VI.

The extremities of a furface are lines.

VII.

A plane furface is that which lies evenly between its ex¬
tremities.

VIII.

A plane angle is the inclination of two lines to one ano¬
ther, in a plane, which meet together, but are not in the
fame direction.

IX.

A plane rectilinear angle is the inclina¬
tion of two ftraight lines to one another,
which meet together, but are not in the
fame ftraight line.

BOOK I. DEFINITIONS .

xix

X.

When one ftraight line handing on ano¬
ther ftraight line makes the adjacent angles
equal, each of thefe angles is called a right
angle , and each of thefe lines is faid to be
perpendicular to the other.

XI.

An obtufe angle is an angle greater
than a right angle.

XII.

An acute angle is an angle lefs than a
right angle.

XIII.

A term or boundary is the extremity of any thing.

XIV.

A figure is a furface enclofed on all fides by a line or lines.

XV.

A circle is a plane figure, bounded
by one continued line, called its cir¬
cumference or periphery; and hay¬
ing a certain point within it, from
which all firaight lines drawn to its
circumference are equal.

XVI.

This point (from which the equal lines are drawn) is
called the centre of the circle.

XX

BOOK I. DEFINITIONS .

XVII.

A diameter of a circle is a ftraight line drawn
through the centre, terminated both ways
in the circumference.

XVIII.

A femicircle is the figure contained by the
diameter, and the part of the circle cut off
by the diameter.

XIX.

A fegment of a circle is a figure contained
by a ftraight line, and the part of the cir¬
cumference which it cuts off.

XX.

A figure contained by ftraight lines only, is called a redti-
linear figure.

XXI.

A triangle is a redtilinear figure included by three fides.

XXII.

A quadrilateral figure is one which is bounded

by four fides. The ftraight lines .'

and «- ■ — connecting the vertices of the

oppofite angles of a quadrilateral figure, are
called its diagonals.

XXIII.

A polygon is a redtilinear figure bounded by more than
four fides.

BOOK I. DEFINITIONS.

xxi

XXIV.

A triangle whofe three lides are equal, is
faid to be equilateral.

XXV.

A triangle which has only two lides equal
is called an ifofceles triangle.

XXVI.

A fcalene triangle is one which has no two lides equal.

XXVII.

A right angled triangle is that which
has a right angle.

XXVIII.

An obtufe angled triangle is that which
has an obtufe angle.

XXIX.

An acute angled triangle is that which
has three acute angles.

XXX.

Of four-lided figures, a fquare is that which
has all its lides equal, and all its angles right
angles.

XXXI.

A rhombus is that which has all its lides
equal, but its angles are not right angles.

XXXII.

An oblong is that which has all its
angles right angles, but has not all its
lides equal.

XXII

BOOK!. POSTULATES .

_XXXIII.

7 A rhomboid is that which has its op-
polite lides equal to one another,
but all its lides are not equal, nor its

angles right angles.

XXXIV.

All other quadrilateral figures are called trapeziums.

XXXV.

Parallel Eraight lines are fuch as are in
__ the fame plane, and which being pro¬

duced continually in both directions,

would never meet.

POSTULATES.

I.

Let it be granted that a Eraight line may be drawn from
any one point to any other point.

II.

Let it be granted that a finite Eraight line may be pro¬
duced to any length in a Eraight line.

III.

Let it be granted that a circle may be defcribed with any
centre at any diEance from that centre.

AXIOMS.

I.

Magnitudes which are equal to the fame are equal to
each other.

II.

If equals be added to equals the fums will be equal.

BOOK I. AXIOMS .

xxm

III.

If equals be taken away from equals the remainders will
be equal.

IV.

If equals be added to unequals the fums will be un¬
equal.

V.

If equals be taken away from unequals the remainders
will be unequal.

VI.

The doubles of the fame or equal magnitudes are equal.

VII.

The halves of the fame or equal magnitudes are equal.

VIII.

Magnitudes which coincide with one another, or exadtly
fill the fame fpace, are equal.

IX.

The whole is greater than its part.

X.

Two ftraight lines cannot include a fpace.

XI.

All right angles are equal.

XII.

If two ftraight lines ( — ' 7 ) meet a third
ftraight line (—■ - ■■ ■ ) fo as to make the two interior

angles ( W and fc ) on the fame fide lefs than
two right angles, thefe two ftraight lines will meet if
they be produced on that fide on which the angles
are lefs than two right angles.

XXIV

BOOK I. ELUCIDATIONS.

The twelfth axiom may be expreffed in any of the fol-

1. Two diverging flraight lines cannot be both parallel
to the fame flraight line.

2. If a flraight line interfedl one of the two parallel
flraight lines it mufl alfo interfedl the other.

3. Only one flraight line can be drawn through a given
point, parallel to a given flraight line.

Geometry has for its principal objects the expofition and
explanation of the properties of figure , and figure is defined
to be the relation which fubfifls between the boundaries of
fpace. Space or magnitude is of three kinds, linear , fiuper-
ficial , and folid.

Angles might properly be confidered as a fourth fpecies
of magnitude. Angular magnitude evidently confifls of
parts, and mufl therefore be admitted to be a fpecies oi
quantity The fludent mufl not fuppofe that the magni¬

tude of an angle is affedled by the length
of the flraight lines which include it, and
of whofe mutual divergence it is the mea-
fure. The vertex of an angle is the point
where the files or the legs of the angle
meet, as A.

An angle is often defignated by a fingle letter when its

legs are the only lines which meet to¬
gether at its vertex. Thus the red and
blue lines form the yellow angle, which
in other fyflems would be called the
angle A. But when more than two

B lines meet in the fame point, it was ne-
ceffary by former methods, in order to
avoid confufion, to employ three letters

E to defignate an angle about that point.

BOOK I. ELUCIDATIONS .

XXV

the letter which marked the vertex of the angle being
always placed in the middle. Thus the black and red lines
meeting together at C, form the blue angle, and has been
ufually denominated the angle FCD or DCF The lines
FC and CD are the legs of the angle; the point C is its
vertex. In like manner the black angle would be delignated
the angle DCB or BCD. The red and blue angles added
together, or the angle HCF added to FCD, make the angle
HCD ; and fo of other angles.

When the legs of an angle are produced or prolonged
beyond its vertex, the angles made by them on both fides
of the vertex are faid to be vertically oppojite to each other :
Thus the red and yellow angles are faid to be vertically
oppofite angles.

Superpojition is the procefs by which one magnitude may
be conceived to be placed upon another, fo as exactly to
cover it, or fo that every part of each fhall exadtly coin¬
cide.

A line is faid to be produced , when it is extended, pro¬
longed, or has its length increafed, and the increafe of
length which it receives is called its produced part , or its
production.

The entire length of the line or lines which enclofe a
figure, is called its perimeter. The firffc fix books of Euclid
treat of plain figures only. A line drawn from the centre
of a circle to its circumference, is called a radius. The
lines which include a figure are called its Jides. That fide
of a right angled triangle, which is oppofite to the right
angle, is called the hypotenufe. An oblong is defined in the
fecond book, and called a reClattgle. All the lines which
are confidered in the firft fix books of the Elements are
fuppofed to be in the fame plane.

The Jlraight-edge and compajfes are the only inftruments,

XXVI

BOOK L ELUCIDATIONS.

the ufe of which is permitted in Euclid, or plain Geometry.
To declare this reftridtion is the objedt of the populates.

The Axioms of geometry are certain general propofitions,
the truth of which is taken to be felf-evident and incapable
of being eftablilhed by demonftration.

Propofitions are thofe refults which are obtained in geo¬
metry by a procefs of reafoning. There are two fpecies of
propofitions in geometry, problems and theorems.

A Problem is a propolition in which fomething is pro-
pofed to be done ; as a line to be drawn under fome given
conditions, a circle to be defcribed, fome figure to be con¬
it rudted, &c.

The folution of the problem conlilts in fhowing how the
thing required may be done by the aid of the rule or Itraight-
edge and compaffes.

The demonfiratton conlilts in proving that the procefs in¬
dicated in the folution really attains the required end.

A Theorem is a propolition in which the truth of fome
principle is afferted. This principle mull; be deduced from
the axioms and definitions, or other truths previously and
independently eflablifhed. To fhow this is the objedt of
demonflration.

A Problem is analogous to a populate.

A Theorem refembles an axiom.

A Pofiulate is a problem, the folution of which is affumed.

An Axiom is a theorem, the truth of which is granted
without demonflration.

A Corollary is an inference deduced immediately from a
propolition.

A Scholium is a note or obfervation on a propolition not
containing an inference of fufficient importance to entitle it
to the name of a corollary.

A Lemma is a propolition merely introduced for the pur-
pofe of eflablifhing fome more important propolition.

XXV11

SYMBOLS AND ABBREVIATIONS.

• •

£

$

+

x

exprelTes the word therefore .

. becaufe.

. equal. This fign of equality may

be read equal to, or is equal to, or are equal to; but
any difcrepancy in regard to the introduction of the
auxiliary verbs is, are, &c. cannot affedt the geometri¬
cal rigour.

means the fame as if the words * not equal ’ were written,
fignifies greater than.

.... lefs than.

.... not greater than.

.... not lefs than .

is read plus (more), the fign of addition ; when interpofed
between two or more magnitudes, fignifies their fum.
is read minus (lefs), fignifies fubtraCtion; and when
placed between two quantities denotes that the latter
is to be taken from the former,
this fign expreffes the produCt of two or more numbers
when placed between them in arithmetic and algebra ;
but in geometry it is generally ufed to exprefs a reffi-
angle, when placed between “ two ffraight lines which
contain one of its right angles.’’ A reffiangle may alfo
be reprefented by placing a point between two of its
conterminous fides.

J l expreffes an analogy or proportion ; thus, if A, B, C
and D, reprefent four magnitudes, and A has to
B the fame ratio that C has to D, the proportion
is thus briefly written,

A : B ;: C : D,

A : B = C : D,

A_ C

° r B D.

This equality or famenefs of ratio is read,

xxviii STMBOLS AND ABBREVIATIONS.

as A is to B, fo is C to D ;
or A is to B, as C is to D.

|| fignifies parallel to.

JL .... perpendicular to.

. angle.

. . right angle,

two right angles.

/K or briefly designates a point.

C, => or flgnifies greater , equal, or lefs than.

The fquare defcribed on a line is concifely written thus,

2

In the fame manner twice the fquare of, is exprefled by

2 • —— * 2 .

def. flgnifies definition.
pos. pofiulate.

ax.

• • •

axiom.

hyp. hypothefis. It may be neceflary here to re¬

mark, that the hypothefis is the condition aflumed or
taken for granted. Thus, the hypothefis of the pro-
pofition given in the Introduction, is that the triangle
is ifofceles, or that its legs are equal,
conft. .... confirudlion. The confirudlion is the change
made in the original figure, by drawing lines, making
angles, defcribing circles, &c. in order to adapt it to
the argument of the demonftration or the folution of
the problem. The conditions under which thefe
changes are made, are as indisputable as thofe con¬
tained in the hypothefis. For inftance, if we make
an angle equal to a given angle, thefe two angles are
equal by conftrudlion.

Q^E. D. Quod erat demonfirandum.

Which was to be demonftrated.

CORRIGENDA .

XXIX

Faults to be correEled before reading this Volume .

Page 13 , line 9, for def. 7 def. 10 .

45, laft li ne 3 for pr. 19 read pr. 29.

54, line 4 from the bottom, for black and red line read blue
and red line.

59, line 4, for add black line fquared read add blue line
fquared.

60, line 17, for red line multiplied by red and yellow line

read red line multiplied by red, blue, and yellow line.
76, line 11, for def. 7 read def. 10.

81, line 10, for take black line read take blue line.

105, line 11, for yellow black angle add blue angle equal red
angle read yellow black angle add blue angle add red
angle.

129, laft line, for circle read triangle.

141, line 1 , for Draw black line read Draw blue line.

196, line 3, before the yellow magnitude infert M.

f

■

, /

.

.

.

<£uclto.

BOOK I.

PROPOSITION I. PROBLEM.

N a given finite
firaight line (——)
to defcribe an equila¬

teral triangle.

r~\

Defcribe I | and

(populate 3.); draw and — (port. 1.)

then will A be equilateral.

— ■" (axiom. 1.) ;

and therefore A is the equilateral triangle required.

Q^E. D.

BOOK I. PROP. II. PROB.

2

ROM a given point ( ■ ' ll"- ),
to draw a firaight line equal
to a given finite firaight

Draw——»— (poft. i.), defcribe

A (pr. i.), produce ■■■> ■— (pod:.

©

2.), defcribe

(poft. 3.), and

(poft. 3.); produce ' (poft. 2.), then

is the line required.

For

(def. 15.),

an( i 1 ■ »— zz 11 > (conft.), .% ■ 1 ■ 11 1

(ax. 3.), but (def. 15.) 1 111 ■■■'■ :

uni—— . drawn from the given point (

is equal the given line

Q. E. D.

BOOK I. PROP. III. PROB.

ROM the greater
(———■•) of

two given Jiraight
lines , to cut off a part equal to
the lefs (•— i).

Draw

(poll:. 3 .), then

(pr. 2.); defcribe

For

and

(def. 15.),
(conft.);

• •

(ax. 1.).

Q. E. D.

4

BOOK I. PROP. IF. THEOR.

F two triangles
have two Jides
of the one
refpeffively
equal to two fdes of the
other , ( n — to ——

the an

•gles ( ▲ and )

contained by thofe equal
fdes alfo equal; then their bafes or their fdes and

) are alfo equal: and the remaining and their remain¬
ing angles oppofte to equal fdes are refpeffiively equal

(>=>44 ) .* and the triangles are
equal in every refpeB.

Let the two triangles be conceived, to be fo placed, that
the vertex of the one of the equal angles, or •

fhall fall upon that of the other, and m ■■ ■■-■■■■■ to coincide
with ■■ i 9 then will m mu coincide with ■ —» « - if ap-

will coincide with

plied: confequently . . .

or two ftraight lines will enclofe a fpace, which is impoffible

(ax. io), therefore

and

9 and as the triangles

coincide, when applied, they are equal in every refpedt.

Q. E. D.

BOOK I. PROP. V. THEOR.

5

N any ifofceles triangle

A

if the equal Jides
he produced, the external
angles at the bafe are equal, and the
internal angles at the bafe are alfo
equal.

Produce

draw

, and

(port. 2.), take

? (p r - 3 -);

■ and

Then in

both, and

— (hyp.) M

and fy = tCy (pr. 4.).

A S. ain /and

in l

M \ and

/y=T\ -

we have

(pr. 4.) but

^ ^ ^ <«• 30

Q. E. D.

6

BOOK I. PROP. FI. THEOR.

and *■
equal .

A

N any triangle ( /. - A ) if

two angles ( and

areequaly theJides (——■■»»»
■** ) ofpojite to them are alfo

For if the tides be not equal, let one
of them ■■■■■ ■— be greater than the

other 9 and from it cut off

■ ■■■■ rz —w 11 mi (pr. 3.), draw

Then in

A- A

(confL) "\

(hyp.) and

common.

the triangles are equal (pr. 4.) a part equal to the whole,
which is abfurd; neither of the fides —— — or
rri—r , ■ is greater than the other, hence they are

equal

E. D.

BOOK I. PROP. VII. THEOR.

7

N the fame bafe («

•), and on

the fame fde of it there cannot be two
triangles having their conterminous
fdes (- 1 1 1 and -'- u ■ 1 1 ,

and in ) at both extremities of

the bafe, equal to each other.

When two triangles Hand on the fame bafe,
and on the fame fide of it, the vertex of the one
fliall either fall outlide of the other triangle, or
within it; or, laftly, on one of its lides.

If it be poffible let the two triangles be con-

ftructed fo that

draw

t=v

and,

(P r - S-)

then

\\

W

and

9

e 9

but (pr. 5.)

therefore the two triangles cannot have their conterminous
fdes equal at both extremities of the bafe.

Q. E. D.

8

BOOK I. PROP. VIII. THEOR.

F two triangles
have two Jides
of the one r effec¬
tively equal to
two fdes of the other

and M .. zz ^

and alfo their bafes (——
»), equal; then the

angles (

and

)

contained by their equalfdes
are alfo equal.

If the equal bafes —— and —— be conceived
to be placed one upon the other, fo that the triangles Ihall
lie at the fame fide of them, and that the equal fides
- and „ . 9 . and . be con¬

terminous, the vertex of the one muft fall on the vertex
of the other; for to fuppofe them not coincident would
contradict the lafi; propofition.

Therefore the fides
cident with

and ,
and

being coin-

9

BOOK I. PROP. IX. PROB. 9

O bifeB a given reBilinear

angle ( 4 >

Take

(P r - 3 -)

draw , upon which

defcribe y (p r - 1 •)»
draw —.

Becaufe ■ zz —■■■■ (conft.)

and —— common to the two triangles

and

(conft.).

A = (P r - 8.)

Q. E. D.

C

10 BOOK I. PROP. X. PROB.

0 bifeffi a given finite Jiraight
line ( 1 —■■■).

Conftrudt

/

/

9 making

4

(pr. i.).

(pr. 9.).

Then

by (pr. 4.),

for

and

(conft.)

4

common to the two triangles.

Therefore the given line is bifedted.

Q. E. D.

BOOK I. PROP. XI. PROB.

( - _

a perpendicular.

ROM a

given

point (—

— )>

in a

given

Jlraight

line

draw

Take any point (-
cut off* . .

) in the given line,
— (P r - 3 -).

conftrudt / \ (pr. i.),

draw and it fhall be perpendicular to

the given line.

For

~ (conft.)
(conft.)

and

common to the two triangles.

Therefore —

(pr. 8.)
(def. io.),

Q^E. D.

12

BOOK I. PROP. XII. PROB .

O draw a
jiraight line
perpendicular
to a given
indefinite Jiraight line
(.■■— ) from a given

[point Ak ) without.

With the given point A\ as centre, at one fide of the
line, and any diftance ■ capable of extending to

the other fide, defcribe

Make
draw —

(pr. io.)

and

then

For (pr. 8.) fince

(conft.)

and

common to both,

= - (def. 15.)

and

• •

(def. 10.).

Q. E. D.

BOOK I. PROP . XIII. THEOR

r 3

HEN a Jlraight line
( ) Jlanding

upon another Jlraight
line ( '■)

makes angles with it; they are
either two right angles or together
equal to two right angles.

If ■ be JL to ■ mi.. ■

then.

(def. 7.),

• *

But if

be not _L to

draw ■ ■ JL . . .— ; (pr. 11.)

Q. E. D.

14

BOOK I. PROP. XIV. THEOR.

F two Jiraight lines

meeting a third jiraight
line ( ■■■■ —rr—w ), at the
fame point , and at oppofite fides of
it, make with it adjacent angles

and equal to

two right angles; thefe ftraight
lines lie in one continuous ftraight
line.

For, if poffible let

and not

be the continuation of

then

+

but by the hypothecs

•’* ^ = ^

+

(ax. 3.); which is abfurd (ax. 9.).

, is not the continuation of

, and

the like may be demonftrated of any other ftraight line
except ■ '-3 ■■ is the continuation

of

BOOK I. PROP. XV. THEOR.

15

In the fame manner it may be fhown that

Q^_E. D.

BOOK I. PROP. XVI. THEOR.

16

F a Jide of a

is produced, the external

an g le ( ) is

greater than either of the
internal remote angles

(

0r A

)•

(pr. io.).

Draw —■ ■ and produce it until

In like manner it can be fhown, that

if

Q. E. D.

BOOK I. PROP. XVII. THEOR.

17

NY two angles of a tri¬

angle j

are to¬

gether lefs than two right angles .

Produce 9 then will

(pr. 16.)

+

and in the fame manner it may be fhown that any other
two angles of the triangle taken together are lefs than two
right angles.

Q. E. D.

D

BOOK I. PROP . XVIII. THEOR.

N any triangle

if one Jide . .. be

greater than another
■—muujuj^ i. 9 the angle op~
pojite to the greater Jide is greater
than the angle oppofite to the lefs.

Make . zz w—if jimh - (pr. 3.), draw

(pr- S-);

(pr. 16.)

and much more

Q. E. D.

BOOK I. PROP. XIX. THEOR.

l 9

F in any triangle

one angle A

A

be greater

than another Hs the Jide
- ii m—w which is oppojite to the greater
angle, is greater than the Jide ■■■ ■
oppojite the lefs.

If

be not greater than

== or "

then mu ft

If

then

▲

(p r - 5 -);

which is contrary to the hypotheiis.

— is not lefs than — ; for if it were,

jMIl zi (P r - l8 -)

which is contrary to the hypotheiis:

• •

Q. E. D..

20

BOOK I. PROP. XX. THEOR .

NY two Jides .- : ,

and m—mm—mm of d

triangle

taken together are greater than the
third Jide (—»).

Produce

>, and

make -1

(p r - 3 -);

draw

Then becaufe

(confl:.),

= 4

c 4

(P r - 5 -)

(ax. 9.)

• •

+

(pr. 19.)

and

+

Q. E. D

BOOK L PROP. XXL THEOR.

21

F from any point ( S )

within a triangle

ftraight lines be
drawn to the extremities of one fde
(—-), thefe lines mufl be toge¬

ther lefs than the other two fides , but
mufl contain a greater angle.

Produce

+

+ - C ■ ■ ■■■■ (pl\ 20.),

add wmmmm to each,

■ I "»■■■■■ j ■

In the fame manner it may be fhown that

- + - 1 = - + -

mmmm “J— LZ “J—

(ax. 4.)

9 • •

which was to be proved.

Again

and alfo

4

• l

(pr. 16.),
(pr. 16.),

4 .

Q^E. D.

22

BOOK I. PROP. XXII. THEOR.

IVEN three right

f mmmmaiBmmmm

lines |

the film of any
two greater than
the third , to conftrudt a tri¬
angle whofe fdes Jhall be re-
fpedhvely equal to the given
lines .

AlTume

Draw

and

(P r - 3 -)-
(pr. 2.).

With

and

as radii.

defcribe

and

(port. 3.);

draw

and

then will

m be the triangle required.

For

and

(conft.)

Q. E. D.

BOOK I. PROP. XXIII. PROB.

23

T a given point ( ) in a

given firaight line (—■■«»),
to make an angle equal to a

given re 51 ilineal angle (

)•

Draw

between any two points

in the legs of the given angle.

Conftruct

JF

(pr. 22.).

fo that

and

Then ^ggfjl == (pr. 8.).

Q. E. D.

24

BOOK I. PROP. XXIV. THEOR.

F two triangles
have two Jides of
the one reflec¬
tively equal to

two fides of the other (.

to .. . - .— ... and -------

to - ), and if one of

the angles ( <3....,,)) contain¬
ed by the equal fides be

greater than the other ( L m V)-> the fide ( —— ) which is
oppofite to the greater angle is greater than the fide ( —)

which is oppofite to the lefs angle.

Make
and 1 ■ ■■

= A

(pr. 23.),
— (P r - 3 -)»

draw

mmmwmm t m

- and

Becaufe

(ax. 1. hyp. conft.)

<p

r. 5.)

but

and /

• ©

but

(pr. 19.)
(pr. 4.)

• •

Q. E. D.

BOOK I. PROP. XXV. THEOR.

2 5

F two triangles
have two Jides

[mmmmmmmmmm an d

.."■■■—) of the

one refpe&tively equal to two

fides (— - and -)

of the other , but their bafes
unequal , the angle fubtended
by the greater bafe ( ■ ■■)

of the one 9 mujl be greater
than the angle fubtended by
the lefs bafe ( ) of the other .

▲

for if

=, Cor Z!

A

= then

14

is not equal to

4

(pr. 4.)

which is contrary to the hypothecs;

4

is not lefs than

for if

4

then

(pr. 24.),

which is alfo contrary to the hypothecs :

... c 4.

E. D.

26

BOOK I. PROP. XXVI. THEOR .

Case I.

F two triangles
have two angles
of the one re-
fpedlively equal
to two angles of the other.

Case II.

),andafde
of the one equal to a fde of
the other fmilarly placed
with refpeB to the equal
angles , the remaining fdes
and angles are refpeBively
equal to one another.

CASE I.

Let »—■ — and » which lie between

the equal angles be equal,
then —" z= * .

For if it be poffible, let one of them .. . ■ be

greater than the other;

4 = A

(pr. 4.)

BOOK I. PROP. XXVI. THEOR.

27

but

and therefore
hence neither of the tides

A

A

(hyp-)

} which is abfurd;
“ and is

greater than the other; and they are equal;

---■» and ^

(pr. 4.).

CASE II.

Again, let

the equal angles

IIMM

“ , which lie oppolite

and | . If it be pofiible, let
9 then take = ■ ■ —— .

draw

Then in

K A

/ -X and i_ we have

• •

but

A= A

and

(P r - 4 -)

(hyp.)

which is abfurd (pr. 16.).

Confequently, neither of the tides or «■—•■***'* is

greater than the other, hence they mull be equal. It
follows (by pr. 4.) that the triangles are equal in all

refpedts.

E. D.

28

BOOK I. PROP. XXVII. THEOR.

Jiraight line
) meet-
two other
Ight lines ,

5 with them the alternate
angles (

are parallel.

) equal, thefe two Jiraight lines

If . . ■ ■ ■■—» be not parallel to

when produced.

they fhall meet

If it be poflible, let thofe lines be not parallel, but meet

when produced; then the external angle \ - is greater

than (pr. 16), but they are alfo equal (hyp.), which

is abfurd : in the fame manner it may be fhown that they
cannot meet on the other fide; they are parallel.

Q. E. D.

BOOK I. PROP. XXVIII. THEOR.

2 9

F aftraight line
(' ), cut¬

ting two other
Jlraight lines
( j l i ilt... and
makes the external equal to
the internal and oppojite
angle , at the fame fide of
the cutting line ( namely ,

, or if it makes the two mternal angles

at the fame fide

together equal to two right angles , thofe two Jlraight lines
are parallel.

II (P r - 2 7 -)

Q^E. D.

3 °

BOOK I. PROP. XXIX. THEOR.

STRAIGHT line

( -.-. ) falling oil

two parallel ftraight
lines ( and

— . ), makes the alternate

angles equal to one another; and
alfo the external equal to the in¬
ternal and oppofte angle on the
fame fde ; and the two internal
angles on the fame fde together
equal to two right angles.

draw

For if the alternate angles ▼ and Mh be not equal,

= i^i (p r - 2 3 )-

■■■■■ ■ ■ (pr. 27.) and there-

■, making

Therefore

fore two ftraight lines which interfedl are parallel to the
fame ftraight line, which is impoflible (ax. 12).

Hence the alternate angles I and are not

unequal, that is, ,he, are equal: ^ ^ if- <S)‘

9 the external angle equal to the inter-

• •

nal and oppofite on the fame fide : if

be added to

both, then

+

(pr-i3)-

That is to fay, the two internal angles at the fame fide of
the cutting line are equal to two right angles.

Q. E. D.

BOOK I. PROP. XXX. THEOR.

TRAIGHT lines ( )

which are parallel to the
fame Jlraight line ( ),

are parallel to one another.

Let

interfed:

Then,

1=1

(pr. 29.),

3 2

BOOK I. PROP. XXXI. PROB.

Draw " ■

make
then ——

ROM a given

point s to
drawnJiraight
line parallel to a given
Jiraight line «■» ).

mmmmm

from the point /

point / to any point

in

= A

(pr. 23*);

(pr. 27.).

Q. E. D.

BOOK I. PROP. XXXII. THEOR.

33

F any Jide (■

)

of a triangle be pro¬
duced, the external

angle ( . \ ) is equal

to the film of the two internal and

oppojite angles ( and M&k ),

and the three internal angles of
every triangle taken together are
equal to two right angles.

Through the point /\ draw

ii

(pr. 31.).

Then

(pr. 29.),

(ax. 2.),

and therefore

+ ^ + A

(p r -13-)-

F

Q. E. D.

34 BOOK I. PROP. XXXIII. THEOR.

TRAIGHT lines ( --

) which join
the adjacent extremities of
two equal and parallelflraight
and «■«»**«»— ),

themfelves equal and parallel.

Draw

the diagonal.

(hyp-)

(pr. 29.)

and

common to the two triangles;

• •

fi

(pr. 4.);

and

11

(pr. 27.).

Q. E. D.

BOOK I. PROP. XXXIV. THEOR.

35

HE oppojite Jides and angles of
any parallelogram are equal ,
and the diagonal ( ■' )

divides it into two equal parts.

Since

▼ = 4

4=1

(pr. 29.)

and ■ common to the two triangles.

Therefore the oppohte lides and angles of the parallelo¬
gram are equal: and as the triangles \ and

are equal in every refpect (pr. 4,), the diagonal divides
the parallelogram into two equal parts.

Q. E. D.

36 BOOK I. PROP. XXXV. THEOR.

ARALLELO GRAMS

on the fame bafe , and
between the fame paral¬
lels , are [in area) equal .

On account of the parallels,

1 (P r - 2 9 -)

_ ^—7 . ’(P r - 2 9 -)

and **

—— — - - J (P r - 34 -)

But,

II

‘T’T

«

CO

\a

minus \ —

\jk

minus

•

• ®

%=

E. D.

BOOK I. PROP. XXXVI. THEOR.

37

ARALLELO-
GRAMS

( ^ it and ) on

equal bafes, and between the
fame parallels, are equal.

Draw

and

> b y (P r - 34 > and hyp.);

= and ||

= and || ——-(pr. 33.)

And therefore

is a parallelogram :

but

(P r - 3S-)

• •

l .

I i A
■ H 1 f

(ax. i.).

\

Q. E. D.

38 BOOK L PROP. XXXVII. THEOR.

RIANGLES

on the fame bafe ( i )

zzW between the fame paral¬
lels are equal.

and

A

Draw

mm m I ■

II

II

(P r - 3 1 -)

Produce

f

and are parallelograms

on the fame bafe, and between the fame parallels,
and therefore equal, (pr. 35.)

Q. E D.

BOOK L PROP. XXXVIII. THEOR.

39

equal bafes and between

the fame parallels are equal.

Draw
and

m w» ssf mm «*

II

II

(pr. 31.).

(pr. 36.);

but

twice (pr. 34.),

and

twice JH (pr. 34.),

(ax. 7.).

Q^E. D.

40

BOOK I. PROP. XXXIX. THEOR.

QUAL triangles

(■

and on the fame bafe

») and on the fame fide of it , are

between the fame parallels.

If ii wn.w« , which joins the vertices
of the triangles, be not || — ,

draw—— || (pr. 31.),

meeting ------- .

Draw 9

Becaufe

(conft.)

(P r - 37 -) :

• &

manner it can

^ , a part equal to the whole,

which is abfurd.

4 f- —; and in the fame

be demonftrated, that no other line except

II

• »

Q. E. D.

BOOK I. PROP. XL. THEOR.

4i

on equal bafes , and on the
fame fide , are between the
fame parallels.

If ■— which joins the vertices of triangles
be not || " - — » ■ ■■■ ,

draw —. || — — ■ ■ (pr. 31.),

meeting -------.

Draw

Becaufe

11

(conft.)

*

_

but

9 a part equal to the whole,
which is abfurd.

"H" ■' J and in the fame manner it

can be demonftrated, that no other line except

—— is II -- - II -.

E. D.

G

42

BOOK I. PROP. XLI. THEOR.

F a paral¬
lelogram

and a triangle
the fame bafe —

■ are upon

' and between
the fame parallels ------ and

——, the parallelogram is double
the triangle.

Draw > . the diagonal ;

Q. E. D.

BOOK I. PROP. XLIL THEOR.

43

O conjlruffi a
parallelogram
equal to a given

triangle
ing an angle equal to a given

rectilinear angle

Make

(pr. io.)

Draw

Make

(P r - 2 3 -)

Draw <

- II

(P r - 3 1 -)

twice

(pr. 41.)

but

(pr. 38.)

1

(£■

1

Q. E. D.

44

BOOK I. PROP. XLIII. THEOR.

the parallelograms which are about
the diagonal of a parallelogram are
equal.

Q. E, D.

BOOK I. PROP. XLIV . PROP.

45

O a given
Jlraight line
( ■" ■ ) to ap¬

ply a parallelo¬
gram equal to a given tri¬

angle (

), and

Ju

having an angle equal to
a given redlilinear angle

( )•

• _

Make zz with

(pr. 42.)

and having one of its ftdes conterminous

with and in continuation of
Produce till it meets || «•■*■*»*»

draw_-- ~, produce it till it meets continued;

draw || ■■■■■» < meeting

produced, and produce

(P r - 43 -J

but

(conft.)

and

▲ (pr. 19. and conft.)

Q. E. D.

46

BOOK I. PROP. XLV. PROS.

O confiruCi a parallelogram equal
to a given rectilinear figure

angle equal to a given rectilinear angle

Draw

and

dividing

the rectilinear figure into triangles.

ConftruCt

having.

Q. E. D.

BOOK I. PROP . XLVI. PROB.

47

fquare .

given Jiraight line
-) to conftruLt a

Draw ' i n ■■■ ■■■ ■ —■ J_ and zz ■■

(pr. 11. and 3.)

Draw 1 || —, and meet¬
ing drawn ||

In

M

(conft.)

= a right angle (conft.)

= = a right angle (pr. 29.),

and the remaining tides and angles muft
be equal, (pr. 34.)

W

is a fquare. (def. 27.)

and Ml

Q. E. D.

48 BOOK I. PROP . XLVII. THEOR .

N 0 njgvfo angled triangle

. the fquare on the

hypotenufe ■ .. . . . ■» is equal to

the fum of the fquares of the fdes, (—■.

and ).

On i i ... , rni—ww and ■

defcribe fquares, (pr. 46.)

DraW mmmmwMwmmm || mmwrnmmmm (pr. 3 I .)

alfo draw 1 ■■■ ■ - and . . .

Again, becaufe

BOOK I. PROP. XLVII. THEOR.

49

In the fame manner it may be fhown

Q E. D.

H

5 °

BOOK I. PROP. XLVIII. THEOR.

the fquare
one Jide

—- ) of

triangle is
equal to the fquares of the
other two fides (nmeip
and —), the angle

) fubtended by that
Jide is a right angle.

Draw _L

and ~

(prs.11.3.)

and draw

alfo.

Since

but —
and

, " 2 +

+

+

and

confequently

(conft.)

2.

2

+

(pr. 47.),
2 (hyp.)

(pr. 8.),

is a right angle.

Q. E. D.

BOOK II.

DEFINITION I.

RECTANGLE or a
right angled parallelo¬
gram is faid to be con¬
tained by any two of its adjacent
or conterminous fides.

Thus: the right angled parallelogram
be contained by the fides -■'■■■■n ■ and

is faid to

or it may be briefly deflgnated by

If the adjacent fldes are equal; i. e. —^
then —““ • '» which is the expreflion

for the redlangle under

and

is a fquare, and

is equal to

or

2

2

or

5 2

BOOK II. DEFINITIONS .

DEFINITION II.

N a parallelogram,
the figure com
of one ot the paral¬
lelograms about the diagonal,
together with the two comple¬
ments, is called a Gnomon.

called Gnomons.

BOOK II. PROP. I. PROB.

53

HE reIIangle contained
by two Jlraight lines ,
one of which is divided
into any number of farts ,

= \ +

(+

is equal to the fum of the rellangles
contained by the undivided line , and the feveral farts of the
divided line .

Draw

and zz

(prs. 2.3. B.i.);

complete the parallelograms, that is to fay,

II

Draw <

i> (pr. 31. B. 1.)

+

II

I

I

+

+

(i; E. D-

54 BOOK II. PROP. II. THEOR.

>

.

if
a

2

*

a

;

i

i

s

*

*

B-

*

a

;

D

8

m m PHVB1S9 ■

F # Jlraight line be divided
into any two parts ^irwH i i— ■
the fquare of the whole line
is equal to the fum of the
|j re 51 angles contained by the whole line and
jj each of its parts .

n

»

n

ii

il

■

!■

tl

a

a

n

a

21

5

s msdsmm im

+

Draw

Defcribe
parallel to

(B. i. pr. 46.)
(B. 1. pr. 31 )

Q. E. D

BOOK II. PROP. III. THEOR.

55

F a Jlraight line be di¬
vided, into any two parts

contained by the whole
line and either of its parts , is equal to
the fquare of that part , together with
the re 51 angle under the parts.

m

Defcribe (pr. 46, B. 1.)

Complete (pr. 31, B. 1.)

In a iimilar manner it may be readily fbown

Q. E. D

BOOK II. PROP. IV. THEOR.

F a fraight line be divided

into any two parts ■ .. mm m m 9

the fquare of the whole line
is equal to the fquares of the
parts, together with twice the rectangle
contained by the parts.

twice ■ • —— .

and

Defcribe
draw •

(pr. 46, B. 1.)

■ (port. 1.),

—

> (pr. 31, B. 1.)

4 =

4 4

(pr. 5, B. 1.),

(pr. 29, B. 1.)

BOOK II. PROP . IV. THEOR.

57

by (prs.6,29, 34. B. 1.)

is a fquare

2

For the fame reafons

P^l is a fquare

1}

w

y

l nr. 43, b. I.)

twice

Q. E. D.

1

BOOK IL PROP. V. PROP .

F a jlraight
line be divided

into two equal
parts andalfo

into two unequal parts ,
the re 51 angle contained by
the unequal parts , together with the fquare of the line between
the points of fedlion, is equal to the fquare of half that line

9 -

59

BOOK II. PROP. F. THEOR.

but

(cor. pr. 4. B. 2.)

and

(conft.)

(ax. 2.)

+

Q. E. D.

6o

BOOK II. PROP . VI. THEOR.

F a fraight line be

and produced to any

the reel angle contained by the
whole line fo increafed, and the
part produced, together with the
fquare of half the line, is equal
to the fquare of the line made up
ofthe half, and the produced part.

Defcribe

(pr. 46, B. 1.), draw

and

i

|| ( ( pr . 31, B. 1.)

but £ = 8 (cor. 4, B. 2.)

Q. E. D.

BOOK II. PROP. VII. THEOR.

61

Defcribe
Draw «

■ ■■■■a m;

7 \r i" 7 ^ j

— (port, i.),

j (pr. 31, B. !.)•

F a Jlraight line be divided
into any two farts 1
the fquares of the whole line
and one of the farts are
equal to twice the redlangle contained by
the whole line and that fart , together
with the fquare of the other farts.

add

to both, (cor. 4, B. 2.)

62

BOOK II. PROP. Fill. THEOR.

Conftrudt

draw

(pr. 46, B. i.);

ii

i H

• ■■■■■MM J

2 _ _ 2 I

(pr. 31, B. 1.)

+ 2

but

+

(pr. 4, B. 11.)

l 2 _

2 •

m •

- +

• •

(pr. 7, B. 11.)
: 4. ..

+

Q. E. D.

BOOK II. PROP. IX. THEOR.

63

F a flraight
line be divided
into two equal
parts !!■■!-- ..
and alfo into two unequal

parts .. - 5 the

fquares of the unequal
parts are together double
thefquares of half the line ,
and of the part between the points of fe 51 ion.

2

2 + 2

Make «. .■ h _L and zz ■—
Draw and

II

or

^ = 4

and draw

hence

(pr. 5, B. 1.) zz half a right angle,
(cor. pr. 32, B. 1.)

(pr. 5, B. 1.) zz half a right angle,
(cor. pr. 32, B. 1.)

■

= ►

zz a right angle.

(prs. 5, 29, B. 1.).
(prs. 6, 34, B. 1.)

I

<

l

1

L

+

— 2 +

or -{-

1

1

(pr. 47, B. I.)

Q. E. D.

64

BOOK II. PROP. X. THEOR .

F a fraight line

. .. . be bi-

fedled and pro¬
duced to any point
—— 9 the fquares of the
whole produced line, and of
the produced part, are toge¬
ther double of the fquares of
the half line, and of the line
made up of the half and pro¬
duced part.

— 2 _j_ 2 — , 1 1imn .

Make

and zz to
draw and

or

" 9

and

II

draw «

> (pr. 31, B. 1.);

alfo.

4

(pr. 5, B. 1.) z= half a right angle,
(cor. pr. 32, B. 1.)

(pr. 5, B. 1.) zz half a right angle
(cor. pr. 32, B. 1.)

4

a right angle.

BOOK II. PROP. X. THEOR.

6 5

half a right angle (prs. 5, 32, 29, 34, B. 1.),

and

9

(p rs - 6, 34, B. 1.). Hence by (pr. 47, B. 1.)

Q. E. D.

K

66

BOOK II. PROP. XI. PROB.

O divide a given Jiraight line —■ m—m
in fuch a manner , that the rectangle
contained by the whole line and one
of its parts may be equal to the
fquare of the other.

_ 9

draw

take

(pr. 46, B. 1.),

■■ (pr. 10, B. 1.),

— (pr- 3. B. 1.)

?

on

defcribe

(pr. 46, B. 1.),

Produce (poft. 2.).

Then, (pr. 6, B. 2.)

+

+

2 •

A e viMiRin

, or,

BOOK II. PROP. XII THEOR.

67

N any obtufe angled
triangle , the fquare
of the fide fubtend-
ing the obtufe angle
exceeds the fium of the fquares
of the fides containing the ob¬
tufe angle , by twice the rec¬
tangle contained by either of
thefie fides and the produced parts
of the fame from the obtufe
angle to the perpendicular let
fall on it from the oppofite acute
angle.

2

2

imam »

.— * +

By pr. 4, B. 2.

-*_|-2

add 8 to both

2 = - 2 (pr. 47, B. 1.)

_ 2

2 •

+

+

^or

+

2 (P r * 47 > 1.). Therefore,

2 . -- . .......... -|- 2

': hence *
by 2

■ • ..........

2 r— 2

4 -

2

Q. E. D.

68

BOOK II. PROP. XIII. THEOR.

FIRST,

SECOND.

N any tri¬
angle , the
fquare op'the
Jidefubt end¬
ing an acute angle , is
lefs than the fum of the
fquares of the fdes con¬
taining that angle , by twice the re 51 angle contained by either
of thefe fdes , and the part of it intercepted between the foot of
the perpendicular let fall on it from the oppofte angle , and the
angular point op' the acute angle.

FIRST.

■— 2 by 2

2

SECOND.

-[- — 2 by 2 •

Firil, fuppofe the perpendicular to fall within the
triangle, then (pr. 7, B. 2.)

add to each - 2 then,

2 -

mi

+ +

(pr. 47, B. 1.)

IIIBB •

" 2 +

2 •

■■■■SB 6

+

BOOK II. PROP. XIII. THEOR.

69

and

+

b y

2 •

Next fuppofe the perpendicular to fall without the
triangle, then (pr. 7, B. 2.)

add to each

2 + — 2 + ■

naaa a •

+

then

2 •

+ -

2 +

2 .

+
2 „

2 •

+

!■■■■ «

by 2

W 71 MBP • mm

7, B. 1.),

4 -

2

■MSgMMHk •

9

maa «

Q. E. D.

7 °

BOOK II. PROP . XIV. PROP.

0 draw a right line of
which the fiquare fhall be
equal to a given recli-
linear figure .

To draw .. ■ ■■— fiuch that ,

Make

produce

take

(pr. io, B. i.).

Defcribe

(port. 3.),

and produce

2 +

to meet it: draw

(■■■■ 9

+

ffiitiir

(pi. ^ t B. 2.),

+

■•••■••a

(pr. 47, B. 1.);

imm a

+

, and

BOOK III.

DEFINITIONS.

I.

QUAL circles are thole whole diameters are
equal.

II.

A right line is said to touch a circle
when it meets the circle, and being
produced does not cut it.

III.

Circles are faid to touch one an¬
other which meet but do not cut
one another.

IV.

Right lines are faid to be equally
difhant from the centre of a circle
when the perpendiculars drawn to
them from the centre are equal.

;\

72

DEFINITIONS.

V.

And the ftraight line on which the greater perpendi¬
cular falls is faid to be farther from the centre.

VI.

A fegment of a circle is the figure contained
by a ftraight line and the part of the circum¬
ference it cuts off.

VII.

An angle in a fegment is the angle con¬
tained by two ftraight lines drawn from any
point in the circumference of the fegment
to the extremities of the ftraight line which
is the bafe of the fegment.

VIII.

An angle is faid to ftand on the part of
the circumference, or the arch, intercepted
between the right lines that contain the angle.

IX.

A fe&or of a circle is the figure contained
by two radii and the arch between them.

DEFINITIONS.

73

X.

Similar fegments of circles
are thofe which contain
equal angles.

Circles which have the fame centre are
called concentric circles.

L

74

BOOK III. PROP. I. PROB.

Draw within the circle any ftraight
line —•««—, make ■■ " ~

draw ■■ ■■ Am ■■ ■ i »

bifedt wmmmmmmm 9 and the point of
bifedtion is the centre.

For, if it be poffible, let any other
point as the point of concourfe of ■
and mmmmmmmmmm be the centre.

zz ------ (hyp. and B. i, def. 15.)

(conft.) and common,

angles; but

(ax. 11.)

which is abfurd; therefore the alfumed point is not the
centre of the circle; and in the fame manner it can be
proved that no other point which is not on is

the centre, therefore the centre is in .. . ■ ■ , and

therefore the point where — ■■■■ !■ — is bifedted is the
centre.

Q. E. D.

BOOK III. PROP. II. THEOR .

75

STRAIGHT line ( ■—— )
joining two joints in the
circumference of a circle

{ ? lies wholly within the circle.

Find the centre of

from the centre draw

to any point in

meeting the circumference from the centre ;
draw and —— .

Then

but

• •

but

• •

• •

= (B. i. pr. 5.)

or

C (B. 1. pr. 16.)

(B. 1. pr. 19.)

• •

every point in

lies within the circle.

Q. E. D.

76 BOOK III. PROP . III. THEOR.

F <2 jlraight line ( ■ . . )

drawn through the centre of a

o

( ■ ■ » ■»»-) which does not pafs through

the centre, it is perpendicular to it; or,
if perpendicular to it, it bifedls it.

bifedls a chord

Draw

and

to the centre of the circle.

In

and

[\

common, and

■< • • iiini

and

Again let

(B. i. pr. 8.)
■■■*>*■ (B. i* def. 7 *)

-L

\n mm*

Then in

and

(B. i. pr. 26.)

and

bifedts

I ■ M *• •

Q. E. D.

BOOK III. PROP. IP. THEOR .

77

another.

F in a circle two Jiraight lines
cut one another , which do
not both pafs through the
centre , they do not bifedl one

If one of the lines pafs through the
centre, it is evident that it cannot be
bifedled by the other, which does not
pafs through the centre.

But if neither of the lines or

pafs through the centre, draw — «■»
from the centre to their interfedlion.

If ' be bife&ed, JL to it (B. 3. pr. 3.)

^ and ^ ■ 1 be

bifedted, _L

( B - 3 - P r - 3 -)

••• * = a

and /. | is J = ^

• a part

equal to the whole, which is abfurd :

•*. " and ■

do not bifedt one another.

Q. E. D.

BOOK HI. PROP. V. THE OR .

F two circles

interfett, they have not the

fame centre.

Suppofe it poffible that two interfe&ing circles have a
common centre; from fuch fuppofed centre draw
to the interfering point, and meeting

the circumferences of the circles.

Then = » (B. i. def. 15.)

and = .— ■■■■■ (B. 1. def. 15.)

/. -- = ————— ; a part

equal to the whole, which is abfurd:
circles fuppofed to interfed in any point cannot
have the fame centre.

Q. E. D.

BOOK III. PROP. VI.

TIIEOR.

79

F two circles

touch

one another internally, they

have not the fame centre.

For, if it be poffible, let both circles have the fame

centre; from fuch a fuppofed centre draw -

cutting both circles, and - - to the point of contact.

Then = —- (B. i. def. 15.)

and = -■■■« — (B. 1. def. 15.)

"""""" = *. . ; a part

equal to the whole, which is abfurd ;
therefore the ahumed point is not the centre of both cir¬
cles ; and in the fame manner it can be demonftrated that
no other point is.

Q E. D.

8q

BOOK III. PROP. VII. THEOR.

FIGURE I.

FIGURE II.

F from any point within a circle

which is not the centre, lines

are drawn to the circumference; the greatef of thofe
lines is that (—■■■■■■) which pajfes through the centre,
and the leaf is the remaining part ( ) of the

diameter.

Of the others , that (■■ .■ m—m ) which is nearer to
the line pafing through the centre, is greater than that
( an r- m mvm ) which is more remote.

Fig. 2. The two lines ( ■ ——— »•»• and ■ .- ■■■ — )

which make equal angles with that pafing through the
centre, on oppofite fides of it, are equal to each other; and
there cannot be drawn a third line equal to them, from
the fame point to the circumference.

FIGURE I.

To the centre of the circle draw — ----- and

then b ■ *"■■■■ nz (B. i. def. 15*)

zz ■ ' — -(- ——— C ■ ■»■■— (B.i. pr. 20.)
in like manner — ^ may be fhewn to be greater than

-- or any other line drawn from the fame point

to the circumference. Again, by (B. 1. pr. 20.)

take from both; ■■ um—« C (ax.).

is lefs

and in like manner it may be fhewn that

BOOK III. PROP. VII. THEOR.

81

than any other line drawn from the fame point to the cir-

— ■ ■■■— C ■■■ (B. i. pr. 24.) and ■— ■

may in like manner be proved greater than any other line
drawn from the fame point to the circumference more
remote from —.

FIGURE II.

If then HZ .. 1 1 1 ? l’f 1101

take - > =z - ■ | 1 11 draw -- 9 then

(B. 1. pr. 4.)

a part equal to the whole, which is abfurd:

‘ — wi»n M« \ and no other line is equal to

■ drawn from the fame point to the circumfer¬
ence ; for if it were nearer to the one palling through the
centre it would be greater, and if it were more remote it
would be lefs.

M

Q. E. D.

82

BOOK III . PPOP. Fill. THFOR.

The original text of this proportion is here divided into
three parts.

I.

F from a point without a circle , ftraight

lines

&c.

are drawn to the cir¬

cumference ; of thofe falling upon the concave circum¬
ference the greateft is that (——■»■») which paft'es
through the centre , and the line ( ) which is

nearer the greateft is greater than that ( ■ i )

which is more remote.

Draw and «**•«*■*•** to the centre.

Then, —which paftes through the centre, is

greateft; for fince --------- ” ------- 9 if

be added to both, ” ' 4“

b ut EZ ■ — (B. i. pr. 20.) —— is greater

than any other line drawn from the fame point to the

concave circumference.

BOOK III. PROP. VIII. THEOR.

83

and

common, but

• •

and in like manner

— (B. 1. pr. 24.);
may be Ihewn C than any

other line more remote from

II.

Of thofe lines falling on the convex circumference the
leaf is that (««--*■») which being produced would
pafs through the centre, and the line which is nearer to
the leaf is lefs than that which is more remote.

For, lince

+

and

(B. 1. pr. 20.)

(ax. 5.)

And again, lince

—| ■ ll«Ma

+

(B. 1. pr. 21.),

mmm 9

■■ And fo of others.

III.

Alfo the lines making equal angles with that which
paf 'es through the centre are equal, whether falling on
the concave or convex circumference ; and no third line
c an be dr awn equal to them from the fame point to the
circumference.

For if -
make

C ■■■■*■ 9 but making —

and draw

84

BOOK III . PROP. Fill. THEOR.

Then in

*

) and /

♦

we have ■>»*■*■»* *****" ■•■■•■■■pi

and

m 1

common, and alfo =

=

but

(B. 1. pr. 4.);

which is abfurd.

■iwiiiih* is not nz

• •

9 nor to any part
is not ZZ

Neither is

»■***!

they are

• •

to each other.

And any other line drawn from the fame point to the
circumference muft lie at the fame fide with one of thefe
lines, and be more or lefs remote than it from the line pall¬
ing through the centre, and cannot therefore be equal to it.

Q. E. D.

BOOK III. PROP. IX. THEOR.

Q

O

F a point be taken within a

more than two equal Jiraight lines

can be drawn to the circumference , that
point muft be the centre of the circle .

For, if it be fuppofed that the point N
in which more than two equal ftraight
lines meet is not the centre, fome other
point — u. muft be; join thefe two points by
and produce it both ways to the circumference.

circle

, from which

Then iince more than two equal ftraight lines are drawn
from a point which is not the centre, to the circumference,
two of them at leaft muft lie at the fame ftde of the diameter

lr»*n»

; and ftnce from a point

A\

which is

not the centre, ftraight lines are drawn to the circumference;
the greateft is — - »■ * , which paftes through the centre :
and »—» ■ which is nearer to

■« tti

but

which is more remote (B. 3. pr. 8.);

1 ” (hyp.) which is abfurd.

The fame may be demonftrated of any other point, dif¬
ferent from f \ which muft be the centre of the circle.

Q. E. D.

Oil

86

BOOK III ; PPOP. X. THEOR.

but as the circles interfedt, they have not the fame

centre (B. 3. pr. 5.):

as . 9 and are drawn

from a point not the centre, they are not equal (B. 3.
prs. 7, 8); but it was fhewn before that they were equal,
which is abfurd; the circles therefore do not interfedt in
three points.

the alfumed point is not the centre of

O ‘ n "* a ° mh,r

more points than two .

For, if it be poffible, let it interfedt in three points;

r\

trom the centre of f, J draw ■ ■ ■ ? —

and ■ ■■■■ -■ ■ ■ to the points of interfedtion ;

(B. 1. def. 15.),"

Q. E. D.

BOOK III. PROP. XI. THEOR.

87

internally , the right line joining their
centres , being produced, Jhallpafs through
a point of contadl.

For, if it be pofhble, let - .

join their centres, and produce it both
ways; from a point of contad: draw

to the centre of

and from the fame point

of contad: draw

to the centre of

Becaufe in

(B. 1. pr. 20.),

I■1B 0 i

and

as they are radii of

88

BOOK III. PROP. XL THEOR.

but

+

J take

away
and -

which is common.

*

but

becaufe they are radii of

O

and CZ a part greater than the

whole, which is abfurd.

The centres are not therefore fo placed, that a line
joining them can pafs through any point but a point of
con tad.

Q. E. D.

BOOK III . PROP. XII ; THEOR.

89

ther externally , the Jiraight line
■ — mi joining their centres,

pajfes through the point of contact.

If it be poffible, let . join the centres, and

not pafs through a point of contact; then from a point of
contact draw and to the centres.

Becaufe ------ -j- —• ■ in .

(B. 1. pr. 20.),

and ■ ■■■'■■ — - zz: (B. 1. def. 15.),

and ... n 1 = (B. 1. def. 15.),

—■ ■■■ -f- — CZZ ■ 9 a part greater

*

than the whole, which is abfurd.

The centres are not therefore fo placed, that fhe line
joining them can pafs through any point but the point of
contact.

Q. E. D.

N

9 °

BOOK III. PROP. XIII. THEOR.

FIGURE I.

FIGURE II.

NE circle can¬
not touch ano¬
ther , either

externally or
internally , in more points
than one.

FIGURE III.

Fig. i. For, if it be poffible, let

and

O

touch one

another internally in two points;
draw —— i joining their cen¬
tres, and produce it until it pafs
through one of the points of contact (B. 3. pr. 11.);
draw .. . and —— ,

But

(B. 1. def. 15.),

/. if ■ ■■ inw- . nr i be added to both,

-+-—;

(B. 1. def, 15.),

* zz .- ; but

.. (B. 1. pr. 20.),

- n ■ — 4*

which is abfurd.

BOOK III. PROP. XIII. THEOR.

91

Fig. 2. But if the points of contact be the extremities
of the right line joining the centres, this ftraight line muft
be bifedled in two different points for the two centres; be-
caufe it is the diameter of both circles, which is abfurd.

Fig. 3. Next, if it be poffible

touch externally in two points; draw —«««—• joining
the centres of the circles, and palling through one of the
points of contact, and draw .. ■ ■ — ■ and ,

and ---<

• •

+

___ -|- .

which is abfurd.

. 1. def. 15.);

(B. 1. def. 15.):

(B. 1. pr. 20.),

There is therefore no cafe in which two circles can
touch one another in two points.

Q E. D.

**•

92

BOOK III. PROP . XIV . THEOR.

QU AL Jlraight lines ( _ )

infcribed in a circle are e-
qually dijl ant from the centre;
and alfo,fraight lines equally
dijiant from the centre are equal .

Then — half (B. 3. pr. 3.)

and ■ ■ ■■■■ »■ (B. 3. pr. 3.)

iince — (hyp.)

and

(B. 1. def. 15.)

• •

but lince

2 ___ _ 2

and

2

wr ¥ IT 0 * m* K

is a right angle
+ —— 2 (B. 1. pr. 47.)
‘ -j- — .1 ■ 2 for the

fame reafon,

0

©

+

*-

+

BOOK III. PROP. XIV. THEOR t

93

« 2

. _ aiHfliinini

m m m m m im c « a

«'•««« vv * v **

Alfo, if the lines —>*«*«*• and mmmmmmm • be

equally diftant from the centre; that is to fay, if the per¬
pendiculars ■BIBQ1IIBH and be given equal, then

11 BS 1 IS

For, as in the preceding cafe,

2 _1_ 2 _ 2|
"7“ •" mummm - .■ ■ —mm, -j-

but

* m m mm91 Mit

2 .

«►

« 6

2 9 and the doubles of thefe

!>■■*> and

1 * m m m m

are alfo equal.

Q. E. D.

94

BOOK III . PROP. XV. THEOR.

FIGURE I.

HE diameter is the great eft Jiraight
line in a circle : and, of all others ,
that which is neareft to the centre is
greater than the more remote.

FIGURE I.

The diameter »n» i is C any line
For draw and

Then z=

but

9

■ (B. i. pr. 20.)

Again, the line which is nearer the centre is greater
than the one more remote.

Firft, let the given lines be

and

winch are at the fame fide of the centre and do

not interfedt;

draw

\

IBUIIBBIBIII

■iRnaaiiHNft

BOOK III. PROP. XV. THEOR.

95

C — (B. i. pr. 24.)

FIGURE II.

Let the given lines be «■ 1 ■ and
which either are at different tides of the centre,
or interfedl; from the centre draw

and L ■■ ' ' and —

make •*«*>»»« ~ .« ? and
draw Hi-. • - —— .

Since — and

the centre, *. .

but - cz

• ammmm
• •

are equally ditiant from
■— (B. 3. pr. 14.);

(Pt. 1. B. 3. pr. 15.),

Q. E. D.

96

BOOK III. PROP. XVI. THEOR.

HE Jiraight
line
drawn
from the
extremity of the diame¬
ter — 1 ■ of a circle
perpendicular to it falls

... without the circle.

»»*

And if any ftraight
line mmmmmrnmm he

drawn from a point
within that perpendi¬
cular to the point of contaB> it cuts the circle.

PART I

If it be poffible, let
again, be _L .

which meets the circle

, and draw

Then, becaufe 9

= ^ ( B - !• P r - S-)»

and each of these angles is acute. (B. 1. pr. 17.)

but = b (hyp.), which is abfurd, therefore

. drawn —. —. — does not meet

the circle again.

BOOK III. PROP. XVI. THEOR .

97

PART II.

Let be 1 1 and let ------ be

drawn from a point % S between and the

circle, which, if it be pofiible, does not cut the circle.

Becaufe

mmmmmummrn

is an acute angle ; fuppofe
■*»■■■ J- drawn from the centre of the

circle, it mull: fall at the fide of the acute angle.

•t> which is fuppofed to be a right angle, is C

m mm mm m » wmmmm ®

but •

•■viflinaii

and mmmmmmm* C ....... ..n, a part greater than

the whole, which is abfurd. Therefore the point does
not fail outfide the circle, and therefore the ftraight line
• ..■•■mi. cuts the circle.

Q. E. D.

o

BOOK III. PROP. XVII. THEOR .

O draw a tangent to a given

given point , either in or outjide of its
circumference.

If the given point be in the cir¬
cumference, as at „„ J , it is plain that
the ftraight line " J_ ---------

the radius, will be the required tan-

gent (B. 3. pr. 16.) But if the given point
outlide of the circumference, draw ■*

circle

from a

from it to the centre, cutting

; and

draW nummmmmmmm JL -------- * defcribe

concentric with

will be the tangent required.

BOOK III. PROP. XVII. THEOR.

99

Q- E. D.

IOO BOOK III. PROP. XVIII. THEOR.

F a right line he

a tangent to a circle , the
Jiraight line drawn

from the centre to the
point of contaffi, is perpendicular to it.

For, if it be poffible,
let ■»*—•••» be JL «■■■■■■»■■»,

then becaufe

is acute (B. i. pr. 17.)

-- c -

(B. 1. pr. 19.);
but Hi 1 ir —- — ?

and

a part greater than

the whole, which is abfurd.

* is not _L . * and in the fame man¬

ner it can be demonftrated, that no other line except
■— is perpendicular to #

Q. E. D.

BOOK III : PROP . Z 7 X

THEOR .

IOI

F Jiraight line _

be a tangent to a circle,
theJiraight line ,

drawn perpendicular to it
from point of the contact, pajfes through
the centre of the circle.

For, if it be poffible, let the centre
be without 9 and draw

• from the fuppofed centre

to the point of contad.

Becaufe »■»•»* *»•*•• J_

(B. 3. pr. 18.)

a part equal to the whole, which is abfurd.

Therefore the affumed point is not the centre; and in
the fame manner it can be demonlhrated, that no other
point without is the centre.

Q. E. D.

102

BOOK III. PROP . XX. THEOR.

FIGURE I

HE angle at the centre of a circle , is double
the angle at the circumference , when they
have the fame part of the circumference for
their bafe.

FIGURE I.

Let the centre of the circle be on

a fide of

Becaufe

k = \

— ^ (B. i. pr. 5.).

or

But

4 = + 5 ,

= twice (B. 1. pr. 32).

FIGURE II.

FIGURE II.

Let the centre be within
circumference; draw

4

? the angle at the
from the angular

point through the centre of the circle;
then , and zzz

becaufe of the equality of the lides (B. 1. pr. 5).

BOOK III. PROP. XX. THEOR. 103

Hence

4

+ * + m +

— twice 4 #

But ^

and

= k + A

• •

twice

4

FIGURE III.

Let the centre be without 4 and

FIGURE III.

draw

Becaufe

zz twice

the diameter.

twice ; and

(cafe 1.);

twice 4.

Q. E. D.

io4 BOOK III. PROP . XXL THEOR.

FIGURE I.

HE angles ( 4,4 ) in the fame

fegment of a circle are equal.

FIGURE I.

Let the fegment be greater than a femicircle, and
draw —— and ——— to the centre.

4

zz twice or twice zz

(B. 3. pr. 20.);

4=4

4

FIGURE II.

FIGURE II.

Let the fegment be a femicircle, 01 lefs than a
femicircle, draw ,1 "'* ,Tr "' 1 the diameter, alfo draw

=4 > = 4

= «4 .

(cafe 1.)

Q. E. D.

BOOK TIL PROP. XXII. THEOR.

10 5

= dh.

Q. E. D.

Draw

the diagonals; and becaufe angles in

the fame fegment are equal zz

r = 4

to both.

• •

two right angles (B. i. pr. 32.). In like manner it may¬
be £hown that.

HE oppofite angles

of any quadrilateral figure in -
fcribed in a circle , are together equal to
two right angles.

p

106 BOOK III. PROP. XXIII. THEOR.

PON the fame
ftraight line,

and upon the
fame fide of it ,
two fimilar fegments of cir¬
cles cannot be confiruBed
which do not coincide.

For if it be poffible, let two fimilar fegments

be conftrudted;

draw any right line
draw -

cutting both the fegments,
and

Becaufe the fegments are fimilar.

(B. 3. def. 10.),

(B. 1. pr. 16.)

which is abfurd : therefore no point in either of
the fegments falls without the other, and
therefore the fegments coincide.

O. E. D.

(V

BOOK III. PROP. XXIV. THEOR.

107

cles upon equal Jlraight
lines (—— and ■■■m ■ )
are each equal to the other.

For, if
that —

be fo applied to
may fall on ——

may be on the extremities

the extremities of
— and

at the fame ftde as

becaufe - m ■■ ■ ■ — ,.. 9

— muft wholly coincide with.. •

and the fimilar fegments being then upon the fame
straight line and at the fame fide of it, muft
alfo coincide (B. 3. pr. 23.), and
are therefore equal.

Q. E. D.

io8

BOOK III. PROP. XXV. PROP.

SEGMENT of a circle
being given , to defcribe the
circle of which it is the
fegment.

From any point in the fegment
draW and ™ bifeft

them, and from the points of bifedtion

draw _L . .

and —i—* _L ——

where they meet is the centre of the circle.

Becaufe _ terminated in the circle is bifedted

perpendicularly by , it palles through the

centre (B. 3. pr. 1.), likewife pahes through

the centre, therefore the centre is in the interfedlion of
thefe perpendiculars.

E. D.

BOOK III. PROP. XXVI. THEOR.

109

N equal circles

the arcs

O w O

on which

Jland equal angles , whether at the centre or circum¬
ference , #r<? equal.

Find, let

draw

at the centre.

and

Then Jtince

0-0

and _have

and

• e

But

O “ ,d o

- (B. 1. pr. 4.).

(B. 3. pr. 20.);

ire limilar (B. 3. def. 10.);

they are alio equal (B. 3. pr. 24.)

no BOOK III. PROP. XXVI. THEOR.

If therefore the equal fegments be taken from the
equal circles, the remaining fegments will be equal;

But if the given equal angles be at the circumference,
it is evident that the angles at the centre, being double
of thofe at the circumference, are alfo equal, and there¬
fore the arcs on which they ftand are equal.

Q. E. D.

BOOK III ; PROP. XXVII. THEOR.

111

the angles and which ft and upon equal

arches are equal , whether they be at the centres or at
the circumferences.

For if it be pofftble, let one of them

be greater than the other
and make

\ = A

• •

•*' = ( B - 3 - P r - 26.)

but (hyp.)

©

A C

a part equal

to the whole, which is abfurd; neither angle
is greater than the other, and
•\ they are equal.

Q. E. D«

I 12

BOOK III. PROP. XXVIII. THEOR.

(B. 3. pi\ 26.)

and

,0-0

(ax. 3.)

Q. E. D.

BOOK III. PROP. XXIX. THEOR.

JI 3

the chords ■ i ■ and --------- which fub-

tend equal arcs are equal.

If the equal arcs be femicircles the proportion is
evident. But if not,

be drawn to the centres;

(B. 3-pr. 27.);

but

and

and <*»»

but thefe are the chords
the equal arcs.

(B. 1. pr. 4.);
fubtending

Q. E. D.

Q

BOOK III.

PROP. XXX. PROB.

114

----- (confL),
is common,

and

(conft.)

8•«■■11889

(B. 1. pr. 4.)

V. (B. 3. pr. 28.),

and therefore the given arc is bifedted,

E. D.

*•*«**

BOOK III. PROP. XXXI. THEOR.

11 5

N a circle the angle in a femicircle is a right
angle , the angle in a fegment greater than a
femicircle is acute , and the angle in a feg¬
ment lefs than a femicircle is obtufe.

FIGURE I.

FIGURE I.

The angle

in a femicircle is a right angle.

Draw

and

and

(B. i. pr. 5.)

the half of two

right angles z= a right angle. (B. 1. pr. 32.)

▲

FIGURE II.

The angle m in a fegment greater than a femi¬
circle is acute.

Draw

the diameter, and

• •

• •

▲

a right angle

is acute.

FIGURE II.

BOOK III . PROP . XXXI. THEOR.

116

FIGURE III.

FIGURE III.

The angle ^ | m a ^ e g ment lefs than femi-
circle is obtufe.

Take in the oppolite circumference any point, to
which draw ■■■—. and

*

Becaufe ^ +

(B. 3. pr. 22.)

=, a

= <S±

but

(part 2.).

is obtufe.

Q. E. D.

BOOK III. PROP. XXXII. THEOR.

117

F a right line .

he a tangent to a circle ,
and from the point of con¬
tain a right line ■ ■ — ■■■ ■
be drawn cutting the circle , the angle

A made by this line with the tangent

is equal to the angle in the alter-

ate fegment of the circle.

If the chord fhould pafs through the centre, it is evi¬
dent the angles are equal, for each of them is a right angle.
(B. 3. prs. 16, 31.)

But if not, draw

from the

point of contadt, it muft pafs through the centre of the
circle, (B. 3. pr. 19.)

(B. 3 .pr. 31.)

+ ^ = f (B. i.pr. 32.)

= (ax.).

Again (ff — /T\ = + | >

(B. 3. pr. 22.)

9 (ax.), which is the angle in
the alternate fegment.

Q. E. D.

n8 BOOK III. PROP. XXXIII. PROB.

N a given Jiraight line
to dejcribe a fegment of a
circle that Jhall contain an
angle equal to a given angle

If the given angle be a right angle,
bifedt the given line, and deicribe a
femicircle on it, this will evidently
contain a right angle. (B. 3. pr. 31.)

If the given angle be acute or ob-
tufe, make with the given line, at its extremity.

, draw

and

make

with

r

defcribe

or

as radius.

for they are equal.

is a tangent to

o

(B. 3. pr. 16.)

divides the circle into two fegments
capable of containing angles equal to

cy and j which were made refpedtively equal

to n and (B. 3-pr. 32.)

Q. E. D.

BOOK III. PROP. XXXIV. PROB.

11 9

O cut off from a given cir¬

cle

a fegment

which jhall contain an angle equal to a
given angle

Draw ' ■■■!■— (B. 3. pr. 17.),

a tangent to the circle at any point;
at the point of contadl make

the given angle ;

contains an angle

the given angle.

Becaufe

and

angle in

is a tangent,
cuts it, the

(B. 3. pr. 32.),

Q. B. D.

120

BOOK III. PROP. XXXV. THEOR.

FIGURE I.

F two chords <

in a circle

interfehl each other , the re hi angle contained
by the fegments of the one is equal to the
re ht angle contained by the fegments of the other.

FIGURE I.

If the given right lines pafs through the centre, they are
bifed:ed in the point of interfe&ion, hence the rectangles
under their fegments are the fquares of their halves, and
are therefore equal.

FIGURE II.

Let

Then

or —

• •

FIGURE II.

not

pafs through the’centre, and
draw ■ ■ — ■ and ■— ■— .

2 (B. 2. pr. 6.),

— X ------ — ,,,a " X

( B * 2 - P r - 50 -

figure iii.

FIGURE III.

Let neither of the given lines pafs through the
centre, draw through their interfe&ion a diameter

and

alfo

x

mmm m • ■

X

(Part. 2.),

x

X

(Part. 2.);

• •

x

x —

Q. E. D.

BOOK III. PROP. XXXVI. THEOR.

121

F from a point without a
circle two ftraight lines be
drawn to it, one of which
— is a tangent to
the circle, and the other — ■ ■■ ■■

cuts it; the reft angle under the whole
cutting line — and the

external fegment 11 is equal to
the fquare of the tangent ■ ■■ ■ ■ .

FIGURE I.

FIGURE I.

Let — pafs through the centre;

draw from the centre to the point of contad ;

minus

or

minus

• •

in • * » ■ i

x

(B. i. pr. 47),

2

.. ?

■ (B. 2. pr. 6).

FIGURE II.

• •

X

FIGURE II.

(B. 3. pr. i8)«
Q. E. D.

\

it

122 BOOK III. PROP. XXXVII. THEOR.

F from a point outfde of a
circle two flraight lines be
drawn , the one —
cutting the circle , the
other meeting it, and if

the re Bangle contained by the whole
cutting line and its ex¬
ternal fegment —.. be equal to

the fquare of the line meeting the circle,
the latter ■ is a tangent to

the circle .

Draw from the given point
mm j a tangent to the circle, and draw from the
centre -- T - ^ ^ and

' ■ ■■ 8 = ——-X--( B - 3 - P r - 3 6 -)

but 2 ~ —X (hyp.),

• •

2

2

J

and

• •

(B. i. pr. 8.);

but

f = 4

a right angle (B. 3. pr. 18.),

and

a right angle,

is a tangent to the circle (B. 3. pr. 16.).

Q. E. D.

BOOK IV.

DEFINITIONS.

I.

RECTILINEAR figure is
faid to be infcribed in another,
when all the angular points
of the infcribed figure are on
the fides of the figure in which it is faid
to be infcribed.

II.

A figure is faid to be defcribed about another figure, when
all the fides of the circumfcribed figure pafs through the
angular points of the other figure.

f

III.

A. rectilinear figure is faid to be
infcribed in a circle, when the vertex
of each angle of the figure is in the
circumference of the circle.

IV.

A rectilinear figure is faid to be cir¬
cumfcribed about a circle, when each of
its fides is a tangent to the circle.

BOOK IF. DEFINITIONS.

124

y.

A circle is faid to be infcribed in
a rectilinear figure, when each fide
of the figure is a tangent to the
circle.

VI.

A circle is faid to be circum-
fcribed about a rectilinear figure,
when the circumference palfes
through the vertex of each
angle of the figure.

is circumfcribed.

VII.

A straight line is faid to be infcribed in
a circle, when its extremities are in the
circumference.

The Fourth Book of the Elements is devoted to the folution of
problems y chiefly relating to the infcriptioh and circumfcrip-
tion of regular polygons and circles.

A regular polygon is one whofe angles and fides are equal.

BOOK IF. PROP. I. PROB.

125

N a given circle

to place a Jlraight line ,
equal to a given Jlraight line (*——),
not greater than the diameter of the
circle.

Draw ««*•
and if

the diameter of

»ft ft• 1

O

9 then

the problem is folved.

But if — — be not equal to

*•« «1

make

- ( h yp-);

(B. i. pr. 3.) with

as radius.

defcribe

draw

O

and

cutting

9 which is the line required.

For

■ »««■ mmm w »

(B. 1. def. 15. conft.)

Q. E. D.

126

BOOK IV. PROP. II. PROB.

N a given circle

O

to m-

fcribe a triangle equiangular
to a given triangle.

To any point of the given circle draw — — . 9 a tangent

(B. 3. pr. 17.); and at the point of contad

make

and in like manner
draw

Becaufe

( B - '■ pr. *3.)

“ (conft.)

(B. 3. pr. 32.)

= If ; alfo

for the fame reafon.

= (B. 1. pr. 32.),

and therefore the triangle infcribed in the circle is equi¬
angular to the given one.

E. D.

BOOK IF. PROP. III. PROB.

127

BOUT a given

circumfcribe a triangle equi¬
angular to a given triangle.

Produce any fide — , of the given triangle both

ways; from the centre of the given circle draw — ■ ■ " " ■ 9
any radius.

Make ^

and

(B. 1. pr. 23.)

At the extremities of the three radii, draw —
— and mmm—mmmm 9 tangents to the
given circle. (B. 3. pr. 17.)

The four angles of

taken together, are

\

equal to four right angles. (B. 1. pr. 32.)

128

BOOK IF. PROP. III. PROB.

but

are right angles (conft.)

two right angles

but

(B. i. pr. 13.)

and

In the fame manner it can be demonftrated that

4 = 4

(B. 1. pr. 32.)

and therefore the triangle circumfcribed about the given
circle is equiangular to the given triangle.

Q. E. D.

BOOK IF. PROP. IF. PROB.

129

N a given triangle

A

to in¬

fer ibe a circle.

Bifedt

(B. 1. pr. 9.)

and »•

from the point where thefe lines

meet draw -

and mmmmm refpedtively per¬
pendicular to ■■

and

In like manner, it may be £hown alfo

hence with any one of thefe lines as radius, deferibe
^ ^ an d it will pafs through the extremities of the

other two; and the fides of the given triangle, being per¬
pendicular to the three radii at their extremities, touch the

circle (B. 3. pr. 16.), which is therefore inferibed in the
given circle.

s

Q. B. D.

130

BOOK IF. PROP. V . PROP.

O deferibe a circle about a given triangle.

Make . zz and - zz

. ( B - !• P r - I®-)

From the points of bifedtion draw —— - and

j_ - and ' refpec-

tively (B. i. pr. u.), and from their point of

concourfe draw ■ , ■■■■■■■■■■■ and-

and deicribe a circle with any one of them, and
it will be the circle required.

common.

(conft.),

■« (B. i. pr. 4.).

In like manner it may be ihown that

therefore a circle deferibed from the concourfe of
thefe three lines with any one of them as a radius
will circumfcribe the given triangle.

Q. E. D.

BOOK IF. PROP. VI. PROB.

N a given circle
infcribe a fquare.

Draw the two diameters of the
circle Am to each other, and draw

and

is a fquare.

For, lince

are, each of them, in

a femicircle, they are right angles (B. 3. pr. 31),

|| ■■ " (B. i.pr. 28):

and in like manner

And becaufe

* ■ Human

(conft.), and
(B. 1. def. 15).
•*. —— = (B. i.pr. 4);

and lince the adjacent fides and angles of the parallelo¬

gram

fquare.

are equal, they are all equal (B. 1. pr. 34);

infcribed in the given circle, is a

Q. E. D.

132

BOOK IF. PROP. VII. PROB .

BOUT a given circle

a fquare.

Draw two diameters of the given
circle perpendicular to each other,
and through their extremities draw

tangents to the circle ;

n

and | is a fquare.

a right angle, (B. 3. pr. 18.)

alfo

(conft.).

11

be demonftrated that
that ' and

in the fame manner it can

and alfo

is a parallelogram, and

becaufe

they are all right angles (B. 1. pr. 34):

it is alfo evident that ' ■“ 9 ? —

and are equal.

is a fquare.

Q. E. D.

BOOK IV. PROP. VIII. PROB.

1 33

Make
and
draw
and ■

9

9

9

(B. i. pr. 31.)

and fince

is a parallelogram;

(hyp.)

• •

is equilateral (B. 1. pr. 34.)

\

In like manner, it can be Ihown that

are equilateral parallelograms;

and therefore if a circle be defcribed from the concourfe
of thele lines with any one of them as radius, it will be
infcribed in the given fquare. (B. 3. pr. 16.)

Q^E. D.

*3 4

BOOK IF. PROP . /X PPOP.

O defcribe a circle about

given fquare

a

D raw the diagonals
and ■ ■ interfering each

other; then.

becaufe

and

have

their iides equal, and the bafe
common to both.

(B. i. pr. 8),

or

is bifeded : in like manner it can be fhown

is bifedted;

hence

their halves,

*. . = *■ ■ 1 ; (B. i. pr. 6.)

and in like manner it can be proved that

If from the confluence of thefe lines with any one of
them as radius, a circle be defcribed, it will circumfcribe
the given fquare.

Q. E. D.

BOOK IV. PROP. X. PROB.

>35

O conflruffi an ifofceles
triangle , in which each of
the angles at the bafe Jhail
he double of the vertical

angle.

Take any ftraight line

and divide it fo that

-x.=

(B. 2. pr. 11.)

With

■■■■■

as radius, defcribe

and place

in it from the extremity of the radius,
(B. 4. pr. 1); draw

V
\ 1 <

Since

Then \ is the required triangle.

For, draw — ■ and delcribe

about A (B. 4. pr. $.)

■ x— =

O

• •

is a tangent to

1 =

O

( B - 3 - P r - 37 ')

(B. 3. pr. 32),

\

i3<3

BOOK IF. PROP. X. PROB.

add 4 to each,

+ 4 = + 4 \

but -j- or 4 — iff' (B. i. pr. 5):

lince

■■■■■■ (B. 1. pr. 5.)

confequently A = ^ + 4 =

(B. 1. pr. 32.)

.*• mmmmrnmamm ZZ .— ■■ (B. I. pr. 6.)

" ■' — .. ZZ: 1 !■■■■■ hi iinm nr (cOIlft.)

4

(B. 1. pr. 5.)

• •

4

A = 4 = = ^3 +

twice and confequently each angle at

the bafe is double of the vertical angle.

Q. E. D.

BOOK IF. PROP. XL PROP.

1 37

N a given circle

to infcribe an equilateral and equi¬
angular pentagon.

Conftrudt an ifofceles triangle, in
which each of the angles at the bafe
fhall be double of the angle at the
vertex, and infcribe in the given

circle a triangle

equiangular to it; (B. 4. pr. 2.)

Bifedt

draw

4 and A (B. 1. pr. 9.)
— . —— , 1 and «-

Becaufe each of the angles

A

▲

A

and \\ are equal,
the arcs upon which they Hand are equal, (B. 3. pr. 26.)

and

and

• -9 . 9 — 9 —-

......... which fubtend thefe arcs are equal (B. 3. pr. 29.)

and .** the pentagon is equilateral, it is alfo equiangular,
as each of its angles ftand upon equal arcs. (B. 3. pr. 27).

Q. E. D.

T

x 3 8

BOOK IF. PROP. XII. PROB.

O defcribe an equilateral
and equiangular penta¬
gon about a given circle

Draw five tangents through the
vertices of the angles of any regular
pentagon infcribed in the given

circle

o (B. 3. pr. 17).

Thefe five tangents will form the required pentagon.

- (B. 1. pr. 47),

and ■ common ;

,.7 =

and

^ = 4 (B

. 1. pr. 8.)

• •

\A

twice

and

4

twice

4

In the fame manner it can be demonftrated that

£1

twice

and

“ twice

but

— (B. 3. pr. 27),

BOOK IV. PROP. XII. PROB.

*39

their halves

= 3, alio J = L

and

common;

• •

and

• •

9

zz twice

In the fame manner it can be demonftrated

that

twice

but

■iMMiaU

9

In the fame manner it can be demonftrated that the
other ftdes are equal, and therefore the pentagon is equi¬
lateral, it is alfo equiangular, for

£S

twice

A.

zz twice

and therefore

| in the fame manner it can be
demonftrated that the other angles of the defcribed

pentagon are equal.

Q. E. D

140

BOOK IV. PROP. XIII. PROB.

O infcribe a circle in a
given equiangular and
equilateral pentagon.

Let O be a given equiangular
and equilateral pentagon ; it is re¬
quired to infcribe a circle in it.

Make zz and zz^P

(B. 1. pr. 9.)

Draw

• ••■ • I IB 9 I

&c.

Becaufe

and

?=4

common to the two triangles

(B. 1. pr. 4.)

And becaufe

zz twice

#= m

hence

twice

is bifedted by

In like manner it may be demonftrated that

&

is

bifedted by •■*«*•«»** 9 and that the remaining angle of
the polygon is bifedted in a limilar manner.

BOOK IF. PROP. XIII. PROB.

H 1

Draw

9 9 See. perpendicular to the

tides of the pentagon.

Then in the two triangles

we have T= A ? (confl.)

and zz:

and

common.

zzz a right angle ;

- . (B. i. pr. 26.)

In the fame way it may be fhown that the five perpen¬
diculars on the tides of the pentagon are equal to one
another.

Defcribe

with any one of the perpendicu¬

lars as radius, and it will be the inferibed circle required.
For if it does not touch the tides of the pentagon, but cut
them, then a line drawn from the extremity at right angles
to the diameter of a circle will fall within the circle, which
has been fhown to be abfurd. (B. 3. pr. 16.)

Q^E. D.

142

BOOK IF. PROP. XIV. PROB.

O defcribe a circle about a
given equilateral and equi¬
angular pentagon.

Bifed

and

by .. and -*-«« . , and

from the point of fedion, draw

■ m •»* mm

and

r =s,

common.

1. pr, 4).

In like manner it may be proved that

therefore

9 and

Therefore if a circle be defcribed from the point where
thefe five lines meet, with any one of them
as a radius, it will circumfcribe
the given pentagon.

Q. E. D.

BOOK IV. PROP. XV. PROS.

of the circles,

triangles, hence

* = ►

are equilateral

= one-third of two right

(B. i. pr. 13);

“ i — = one-third of

(B. 1. pr. 32), and the angles vertically oppofite to thefe
are all equal to one another (B. 1. pr. 15), and ftand on
equal arches (B. 3. pr. 26), which are fubtended by equal
chords (B. 3. pr. 29); and hnce each of the angles of the
hexagon is double of the angle of an equilateral triangle,
it is alfo equiangular. Q^_ E D

through its centre, and draw the diameters
9 ■ and 1 ™ * draw

mmmmmmmmm ^ —« —“ ? ......... ? &C. and the

required hexagon is infcribed in the given
circle.

Since

pafles through the centres

O infcribe an equilateral and equian¬
gular hexagon in a given circle

o

From any point in the circumference of

the given circle defcribe

palling

r 44

BOOK IF. PROP. XVI. PROP .

O infcribe an equilateral and
equiangular quindecagon in
a given circle .

Let ——. and be

the fides of an equilateral pentagon
infcribed in the given circle, and
■ - — - the fide of an inscribed equi¬
lateral triangle.

The arc fubtended by
. and i

of the whole
circumference.

The arc fubtended by

of the whole
circumference.

Their difference zz fr

the arc fubtended by ———— zz T V difference of
the whole circumference.

Hence if flraight lines equal to be placed in the

circle (B. 4. pr. 1), an equilateral and equiangular quin¬
decagon will be thus infcribed in the circle.

Q. E. D.

BOOK V.

DEFINITIONS.

I.

LESS magnitude is faid to be an aliquot part or
fubmultiple of a greater magnitude, when the
lefs meafures the greater; that is, when the
lefs is contained a certain number of times ex¬
actly in the greater.

II.

A greater magnitude is faid to be a multiple of a lefs,

when the greater is meafured by the lefs; that is, when

the greater contains the lefs a certain number of times \\

exactly.

III.

Ratio is the relation which one quantity bears to another
of the fame kind, with refpedt to magnitude.

IV.

Magnitudes are faid to have a ratio to one another, when
they are of the fame kind; and the one which is not the
greater can be multiplied fo as to exceed the other.

The other definitions will be given throughout the book
where their aid is firfi required .

u

146

AXIOMS.

I.

QUIMULTIPLES or equifubmultiples of the
fame, or of equal magnitudes, are equal.

If A n: B, then
twice A zzz twice B, that is.

2 A — 2 B;

3 A = 3 B;

4 A = 4 B ;

&c. &c.

and i of A zz \ of B ;
\ of A zz i of B ;

&c. &c.

II.

A multiple of a greater magnitude is greater than the fame
multiple of a lefs.

Let A CZ B, then
2AC2B;

3 A CZ 3 B ;

4AC4B;

&c. &c.

III.

That magnitude, of which a multiple is greater than the
fame multiple of another, is greater than the other.

Let 2 A SZ 2 B, then
A C B;

or, let 3 A CZ 3 B, then
ACZB;

or, let m A IZ m B, then
A CZ B.

BOOK V. PROP . I. THEOR ,

147

F any number of magnitudes be equimultiples of as
many others , each of each : what multiple soever
any one of the fir ft is of its part , the fame multiple
Jhall of the firft magnitudes taken together be of all

the others taken together .

Let QQQQQ be the fame multiple of Q 5
that is of

that OOOOO O*

Then is evident that

QQQQQ

> is the fame multiple of <

OOOOO J

which that QQQQQ isofQ •

becaufe there are as many magnitudes

,Q

in

Q

V

o

QQQQQ '

VIV99

^ <

, OOOOO J

k.

o

as there are in QQQQQ = Q.

The fame demonftration holds in any number of mag¬
nitudes,, which has here been applied to three.

If any number of magnitudes, &c.

148

BOOK V. PROP. II. THEOR.

F the firft magnitude be the fame multiple of the
fecond that the third is of the fourth, and the fifth
the fame ??mltiple of the fecond that the fixth is oj
the fourth, then fhall the fir ft, together with the
fifth, be the fame multiple of the fecond that the third, together
with the fixth, is of the fourth .

Let 9 | 1 , the firft, be the fame multiple of } ?
the fecond, that 666, the third, is of < 3 , the fourth;
and let ) ; J , the fifth, be the fame multiple of ,

the fecond, that COCO, the fixth, is of O, the
fourth.

Then it is evident, that <

J

, the firft and

fifth together, is the fame multiple of £ > t ^ e fecond,

6>00
OOOO

the fame multiple of the fourth ; becaufe there are as

that

, the third and fixth together, is of

many magnitudes in

1

J

as there are

[ OOO 1

looooj

e

© ®

If the firft magnitude, &c.

BOOK V. PROP. III. rHEOR.

H9

F the firfl of four magnitudes be the fame multiple
of the fecond that the third is of the fourth, and
if any equimultiples whatever of the firfl and third
be taken, thofe Jhall be equimultiples ; one of the

fecond, and the other of the fourth.

The First. The Second.

which

The Third. The Fourth.

of ;

take <;

r

> the fame multiple of ■<

which

♦♦♦♦

,

]♦♦♦♦

r

. r I

> is of -<

I

L

♦♦

♦♦

Then it is evident.

that *<

The Second,

> is the fame multiple of

I

i5°

BOOK V . PROP . ///. THEOR.

which <

♦♦♦♦
♦♦♦♦
♦♦♦♦

The Fourth.

is of i ? ;

becaufe <

y contains <

y contains

as many times as

♦♦♦♦

♦♦♦♦

>■ contains <

♦♦
♦ ♦

contains |

The fame reafoning is applicable in all cafes.
If the firft four, &c.

BOOK V. DEFINITION V.

151

DEFINITION V.

Four magnitudes, 0 , g *, , 0, are faid to be propor¬

tionals when every equimultiple of the firft and third be
taken, and every equimultiple of the fecond and fourth, as.

of the firft d 0

of the third >

•••

♦♦♦

••••

♦♦♦♦

♦♦♦♦♦

••••••

♦♦♦♦♦♦

See.

See.

of the fecond

of the fourth ^ W

■■■

fff

■■■■

ffff

Mfff

■■■■■■

ffffff

See.

&c.

r

Then taking every pair of equimultiples of the firft and
third, and every pair of equimultiples of the fecond and
fourth,

C’ = or □

C=> = or Z 3

= or 3

EZ> = or 33
I , = or —I

If <

♦ ♦ c*

♦ ♦ t=.

then will < ♦♦ c.

♦♦

♦ ♦ t=.

or Z1

or H

or Zl

or

or ;□

1 5 2

BOOK V. DEFINITION V.

That is, if twice the firft be greater, equal, or lefs than
twice the fecond, twice the third will be greater, equal, or
lefs than twice the fourth ; or, if twice the firft be greater,
equal, or lefs than three times the fecond, twice the third
will be greater, equal, or lefs than three times the fourth,
and so on, as above exprefied.

If <

then

will

• •• C, = or Z3

■■

WWW C, = or m

■■■

• •• C, = or □

■■■■

• ••

• •• C, = or =]

See.

&C.

444 t=, = or Zl

ww

<: C, = or Zl

WWW

444 c, = or Zl

wwww

< C, = or Zl

wwwww

c, = or ::-2

i-::r- V -VFp V

&cc.

&c.

In other terms, if three times the firft be greater, equal,
or lefs than twice the fecond, three times the third will be
greater, equal, or lefs than twice the fourth; or, if three
times the firft be greater, equal, or lefs than three times the
fecond, then will three times the third be greater, equal, or
lefs than three times the fourth; or if three times the firft
be greater, equal, or lefs than four times the fecond, then
will three times the third be greater, equal, or lefs than four
times the fourth, and so on. Again,

BOOK V. DEFINITION V.

*53

I ••••

&c.

C, = or Z]

C, = or ZI

C, “ or 3 ■■■■

EZ, = or !□ BHPH

C, z: or □ ■■■■

&c.

&c.

And so on, with any other equimultiples of the four
magnitudes, taken in the fame manner.

Euclid exprefies this definition as follows :—

The firft of four magnitudes is faid to have the fame
ratio to the fecond, which the third has to the fourth,
when any equimultiples whatfoever of the firft and third
being taken, and any equimultiples whatfoever of the
fecond and fourth ; if the multiple of the firft be lefs than
that of the fecond, the multiple of the third is alfo lefs than
that of the fourth; or, if the multiple of the firft be equal
to that of the fecond, the multiple of the third is alfo equal
to that of the fourth; or, ir the multiple of the firft be
greater than that of the fecond, the multiple of the third
is alfo greater than that of the fourth.

In future we fhall exprefs this definition generally, thus :

If M • C, = or Z 2 « 1 ,
when M ; v C, = or m ^

x

1 54-

BOOK V. DEFINITION V.

Then we infer that £ 9 the firft, has the fame ratio

to 3 the fecond, which 4 ,- , the third, has to ^ the
fourth : exprefied in the fucceeding demonftrations thus:

• •
• •

or thus.

♦ = *

or thus, — zz —■ : and is read.

(t

as

is to

so is

♦ to*

?5

And if 9 : | :: 4 we ^ n ^ er ^

M $ C, zz or ^ //z ^ , then will
M C, =: or □

That is, if the firft be to the fecond, as the third is to the
fourth; then if M times the firft be greater than, equal to,
or lefs than m times the fecond, then {hall M times the
third be greater than, equal to, or lefs than m times the
fourth, in which M and m are not to be confidered parti¬
cular multiples, but every pair of multiples whatever;

nor are fuch marks as $, £ , &c. to be confidered

any more than reprefentatives of geometrical magnitudes.

The iludent fhould thoroughly underftand this definition
before proceeding further.

BOOK V. PROP . IF. THEOR.

l 5 S

F the firji of four magnitudes have the fame ratio to
the fecond, which the third has to the fourth, then
any equimultiples whatever of the firft and third
shall have the fame ratio to any equimultiples of
the fecond and fourth ; viz., the equimultiple of the firf fall
have the fame ratio to that of the fecond, which the equi¬
multiple of the third has to that of the fourth .

Let ) : £ :: 4^ : 7 , then 3 : 2 p|:: 3 p : 2 _ ,

every equimultiple of 3 | and 3 ♦ are equimultiples of
and ^ , and every equimultiple of 2 H and 2 , are

equimultiples of ^ and (B. 5, pr. 3.)

That is, M times 3 and M times 3 ^ are equimulti¬
ples of 7^ and + > and m times 2 p| and m 2 : are equi¬
multiples of 2 p| and 2 ; but : :: + ;

(hyp); if M 3 C, =, or □ « 2 then
M3 + C, =, or □ ^ 2 (def. 5.)

and therefore 3 » : 2 H IJ 3 ♦ : 2 7 (def. 5.)

The fame reafoning holds good if any other equimul¬
tiple of the firft and third be taken, any other equimultiple
of the fecond and fourth.

If the firft four magnitudes, &c.

*5 6

BOOK V. PROP . V. THEOR .

F 07 Z£- magnitude be the fame multiple of another ,
which a magnitude taken from the firfl is of a mag¬
nitude taken from the other , the remainder fhall be
the fame multiple of the remainder, that the whole

is of the whole.

Q

Let < 3 >Q = M'

□

o

QQ minus = M' u minus M' m,

o

= M'( A minus .),

OQ ^

If one magnitude, &c.

BOOK V. PROP. VI. THEOR.

1 57

F two magnitudes be equimultiples of two others ,
and if equimultiples of thefe be taken from the firjl
two y the remainders are either equal to thefe others ,
or equimultiples of them.

Q

Let M' ■ ; and O O = M' a ;

o

then OO minus rri m zz

O

M' m minus rri m ~ (M' minus rri) «,

and □o minus rri i z M' a minus rri l =

(M' minus rri) t, .

Hence, (M' minus rri) m and (M' minus rri) a are equi¬
multiples of A and ▲ , and equal to ■ and a 9
when M' minus rri zz \.

If two magnitudes be equimultiples, &c.

i 5 8

BOOK V. PROP. A. THEOR.

F the jirjl oj the four magnitudes has the fame ratio
to the fecond which the third has to the fourth,
then if the firf be greater than the fecond, the
third is alfo greater than the fourth ; and if equal,
equal; if lefs, lefs.

Let : | • T I ; therefore, by the fifth defini¬
tion, if £ C H, then will i= > ;

but if • c ■ , then •• C ■■
and V* C ■> ,

and Cl

Similarly, if 3| =, or then will z=,

or Z]^.

If the firfi: of four, &c.

DEFINITION XIV.

Geometricians make ufe of the technical term “ Inver-
tendo,” by inverfion, when there are four proportionals,
and it is inferred, that the fecond is to the firfi; as the fourth
to the third.

Let A : B :: C : 1 , then, by “ invertendo” it is inferred
B : A :: : C .

BOOKF. PROP. B. PHEOR. 159

F four magnitudes are proportionals , they are pro¬
portionals alfo when taken inverfely.

Let ,

then, inverfely, Q : :: :

If M th en M 9 Z 1 m

by the fifth definition.

Let M Urn that is, m □ CM ,

M 1 □ m 9 or, m CM ;

/. if m O CM , then will m CM 1 ,

In the fame manner it may be fhown,

that if m ^ zz or Z 3 M ,
then will m =, or □ M 8 ;
and therefore, by the fifth definition, we infer
that Q : : .

/. If four magnitudes, &c.

i6o

BOOK V. PROP. C. THEOR.

F the fir ft be the fame multiple of the fiecond , or the
fame part of it, that the third is of the fourth ;
the firfi is to the fecond, as the third is to the
fourth.

Let . the firft, be the fame multiple of the fecond,

that ♦ ♦ , the third, is of ifc, the fourth.

Then

♦ ♦
♦ ♦

take M

becaufe

m

♦♦

m

is the fame multiple of

that is of A (according to the hypothefis);

and M is taken the fame multiple of

that M 'X is of ,
(according to the third propofition).

M

is the fame multiple of

that M

is of

BOOK V : PROP. C. THEOR.

Therefore, if M

16 r

be of fa a greater multiple than

♦♦

m fa is, then M is a greater multiple of fa than

//z A is; that is, if M

be greater than m fa 9 then

M

♦ ♦
♦ ♦

will be greater than m fa • in the fame manner

it can be fhewn, if M

be equal m fa, then

M will be equal m fa .

♦ ♦

And, generally, if M

C, = or

m

then M will be CZ, = or —l m

by the fifth definition,

..

Next, let £ be the fame part of

a

that A is of ^ ^

In this cafe alfo

• •
• •

For, becaufe

is the fame part of

that

♦♦

♦♦

is of ^ ^

162

BOOK V. PROP . C. THEOR.

therefore

is the fame multiple of

that ^ ^ is of £

Therefore, by the preceding cafe,

♦ ♦

and

♦♦

by proportion B.

If the fird: be the fame multiple, &c.

BOOK V. PROP. D. THEOR.

163

F the firjl be to the fecond as the third, to the fourth ,
and if the JirJi be a multiple , or a part of the
fecond; the third is the fame multiple , or the fame
part of the fourth.

and firfb, let

be a multiple

^ ^ fhall be the fame multiple of ?

First. Second. Third. Fourth.

♦ ♦ .

♦ ♦ *

a 00

DQ OO

Take - —

QO

Whatever multiple
take

is of

the fame multiple of

then, becaufe : jj :: ^ ^ : O

and of the fecond and fourth, we have taken equimultiples,

and

OO

, therefore (B. 5. pr. 4),

BOOK V. PROP. D. THEOR.

164

. □ .OO

■QQ ’OO

, but (conft.).

_ □ . (B . pr A) 44 _ OO

- QQ • • 5 - A.) ^

and is the fame multiple of

that

is of

Next, let

m

♦♦

♦♦

and alfo H a part of

then fhall be the fame part of ^ .

Inverfely (B. 5.),

H , -mt
♦♦ ■ '•

but | is a part of

that is.

is a multiple of | ;

♦ ♦

by the preceding cafe, | ^ is the fame multiple of ^
that is, C/ is the fame part of

that ■ is of

&

<1 *

If the hr ft be to the fecond, &c.

BOOK V. PROP . VII. THEOR

165

JUAL magnitudes have the fame ratio to the fame
magnitude , and the fame has the fame ratio to equal
magnitudes .

Let 0 — and any other magnitude ;

then % : ■ = t : ■ an d PS : # = ■ : ^ ■

Becaufe — # ,

M • = M 4 ;

if M ^ C. = or I m 9 • then

M C» == or m ] ,
and \ = + : I (B. 5. def. 5).

From the foregoing reafoning it is evident that,
if m | C. = or Z1 M • , then

m C. = or Z3 M +

| : # = : t; (B. j. def. 5).

/. Equal magnitudes, &c.

166

BOOK V. DEFINITION VII.

DEFINITION VII.

When of the equimultiples of four magnitudes (taken as in
the fifth definition), the multiple of the firft is greater than
that of the fecond, but the multiple of the third is not
greater than the multiple of the fourth; then the firft is
ikid to have to the fecond a greater ratio than the third
magnitude has to the fourth: and, on the contrary, the
third is faid to have to the fourth a lefs ratio than the firft
has to the fecond.

If, among the equimultiples of four magnitudes, com¬
pared as in the fifth definition, we fhould find

• •••• C= llll , but

= or ^ or if we fhould

find any particular multiple M' of the firft and third, and
a particular multiple m of the fecond and fourth, fuch,
that M' times the fir ft is HZ m times the fecond, but M'
times the third is not C ni times the fourth, /. e. z= or
3 ] m times the fourth; then the firft is faid to have to
the fecond a greater ratio than the third has to the fourth;
or the third has to the fourth, under fuch circumftances, a
lefs ratio than the firft has to the fecond : although feveral
other equimultiples may tend to fhow that the four mag¬
nitudes are proportionals.

This definition will in future be expreffed thus:—

If M' ^ C ni Q, but M' = or ;□ ni ,
then •

In the above general exprellion, M and ni are to be
confidered particular multiples, not like the multiples M

BOOK V. DEFINITION FII.

167

and m introduced in the fifth definition, which are in that
definition confidered to be every pair of multiples that can

be taken. It muft alfo be here obferved, that ^ , D,U,
and the like fymbols are to be confidered merely the repre-
fentatives of geometrical magnitudes.

In a partial arithmetical way, this may be fet forth as
follows :

Let us take the four numbers, 8. 7, io, and

Firji .

Second.

Third.

Fourth.

8

7

10

9

16

H

20

18

24

21

3 °

27

32

28

4 °

36

40

35

5 °

4 ?

48

42

60

54

5 6

49

70

y

6 3

6 4

56

80

72

*7 2

63

90

01

80

70

100

90

88

V

no

99

96

84

120

108

104

91

130

117

112

98

140

126

6cc.

See.

See

&c.

Among the above multiples we find 16 C 14 and 20
C that is, twice the firfi is greater than twice the
fecond, and twice the third is greater than twice the fourth;
and 16 3] 21 and 20 ^ that is, twice the firfi: is lefs
than three times the fecond, and twice the third is lefs than
three times the fourth; and among the fame multiples we
can find 71 C 56 and C that is, 9 times the firfi:
is greater than 8 times the fecond, and 9 times the third is
greater than 8 times the fourth. Many other equimul-

168

BOOK V. DEFINITION VII.

tiples might be selected, which would tend to fhow that
the numbers 8, 7, - were proportionals, but they are

not, for we can find a multiple of the firfi: [Z a multiple of
the fecond, but the fame multiple of the third that has been
taken of the firfi; not EZ the fame multiple of the fourth
which has been taken of the fecond; for inftance, 9 times
the firfi; is Q 10 times the fecond, but 9 times the third is
not Q 10 times the fourth, that is, 72 [Z 70, but 90
not CZ or 8 times the firfi; we find IZ 9 times the
fecond, but 8 times the third is not greater than 9 times
the fourth, that is, 64 |Z 63, but So is not £Z When
any fuch multiples as thefe can be found, the firfi; (8) is
faid to have to the fecond (7) a greater ratio than the third
(1 o) has to the fourth and on the contrary the third
(10) is faid to have to the fourth a lefs ratio than the
firfi: (8) has to the fecond (7).

BOOK V . PROP . r/Z7. THEOR.

F unequal magnitudes the greater has a greater
ratio to the fame than the lefs has: and the fame
magnitude has a greater ? atio to the lefs than it
has to the greater.

k

and 11 be two unequal magnitudes,
and any other.

▲

We £hail firfl prove that d which is the greater of the
two unequal magnitudes, has a greater ratio to f;. than d,
the lefs, has to d *

▲

that is, I : Q c : 0 ;

▲

take M' d a ni d , M' , and ml d *
fuch, that M' k and M' d fhall be each Cl | } j
alfo take m If the lead; multiple of d,
which will make m Q CM' M' ;

/. M' is not C ni Q.,

k

but M' J| is C m ' j, y for,

as ml d is the firfl multiple which firfl becomes F M'J| .
than [ni minus i) d or m ff minus ff is not Cl M' d.

and d is not I M' a.

rrl J minus mull be □ M' | -}- M' ▲ ;

▲

that is, ni d mufl be I M' d :

▲

/. M' (| is CZ ni | , but it has been fhown above that

z

BOOK V. PROP. VIII. THEOR.

170

M' is not Cl m ' 5 therefore, by the feventh definition,

▲

| has to a greater ratio than j :

Next we fhall prove that has a greater ratio to _ , the

▲

lefs, than it has to £, the greater;

▲

or, f c :

A

Take rri (), M' J| ? rri , and M' ,
the fame as in the firfi cafe, fuch, that
M' a and M' || will be each C # , and rri £ the lead;
multiple of , which firfi: becomes greater
than M' £ = M' | .

/, rri minus is not^Z M'
and is not C M' A ; confequently

rri minus + is HI M' |: + M' A ;

A

/. rri is Z 2 M' ■, and by the feventh definition,

A

0 has to a greater ratio than has to £.

Of unequal magnitudes, &c.

The contrivance employed in this propofition for finding
the multiples taken, as in the fifth definition, a mul~
tiple of the firfi: greater than the multiple of the fecond, but
the fame multiple of the third which has been taken of the
firfi:, not greater than the fame multiple of the fourth which
has been taken of the fecond, may be illuftrated numerically

as follows :—

The number 9 has a greater ratio to 7 than has to :
that is, 9 : 7 C : 7 i or > 8 + 1 : C : •

BOOK V. PROP. VIII. THEOR,

171

The multiple of 1, which firft becomes greater than,
is 8 times, therefore we may multiply the firft and third
by 8, 9, 10, or any other greater number; in this cafe, let
us multiply the firft and third by 8, and we have ^ 4 -j- 8
and : again, the firft multiple of which becomes
greater than 64 is 10 times; then, by multiplying the
fecond and fourth by 10, we £hall have jo and ; then,
arranging thefe multiples, we have—

8 times 10 times 8 times xo times

the first. the second. the third. the fourth.

64+ 8 70 64 JQ

Confequently 64 -J- 8, or 7 2, is greater than 70 , but
is not greater than to , .*. by the feventh definition, 9 has a
greater ratio to 7 than has to 7 .

The above is merely illuftrative of the foregoing demon-
ftration, for this property could be ftiown of thefe or other
numbers very readily in the following manner; becaufe, if
an antecedent contains its confequent a greater number of
times than another antecedent contains its confequent, or
when a fraction is formed of an antecedent for the nu¬
merator, and its confequent for the denominator be greater
than another fraction which is formed of another antece¬
dent for the numerator and its confequent for the denomi¬
nator, the ratio of the firft antecedent to its confequent is
greater than the ratio of the laft antecedent to its confe¬
quent.

Thus, the number 9 has a greater ratio to 7, than 8 ha!
to 7, for | is greater than

Again, 17 : 19 is a greater ratio than 13 : 15, becaufe

17

19

l 7 X 15 — 2 _^5 onrl 13 _ 13 X 19 _ 247 ^

19 x 15 — 285’ and TE — iTx 19 — hence u 11

x 9 19 X 15 2«5' i 5 — 15 x 19 — 285’ AL Ai

evident that gj? is greater than !g, U is greater that

♦

BOOK V. PROP. VIII. THEOR.

172

1 ^ •

and, according to what has been above fhown, 17 has
to 19 a greater ratio than 13 has to 15.

So that the general terms upon which a greater, equal,
or lefs ratio exifts are as follows:—

A c

If g be greater than jj, A is faid to have to B a greater

ratio than C has to D ; if — be equal to then A has to

B the fame ratio which C has to D ; and if ^ be lefs than
c

A is faid to have to B a lefs ratio than C has to D.

The fludent ffiould underftand all up to this proportion
perfectly before proceeding further, in order fully to com¬
prehend the following proportions of this book. We there¬
fore ftrongly recommend the learner to commence again,
and read up to this dowly, and carefully reafon at each ftep,
as he proceeds, particularly guarding againd: the mifchiev-
ous fyftem of depending wholly on the memory. By fol¬
lowing thefe inftrudlions, he will dnd that the parts which
ufually prefent condderable difficulties will prefent no diffi¬
culties whatever, in profecuting the ftudy of this important
book.

BOOK V ; PROP . /X PHEOR.

J 73

AGNITUDES which have the fame ratio to the
fame magnitude are equal to one another; and
thofe to which the fame magnitude has the fame
ratio are equal to one another .

Let * | : : % * '_ ? then = $ .

For, if not, let C ( ' ? then will

f : C • (B. 5. pr. 8),

which is abfurd according to the hypothecs.

is not CZ . > .

In the fame manner it may be fhown, that

£ is not Cl ,

® ©

♦ =#•

Again, let | : : : 3 A ? then will

For (invert.) q : :: £ :

therefore, by the firft cafe, —

Magnitudes which have the fame ratio, &c,

This may be fhown otherwife, as follows:—

Let : B — C, then B — C, for, as the fraction
— = the fraction —, and the numerator of one equal to the
numerator of the other, therefore the denominator of thefe
fractions are equal, that is B z= C.

Again, if B : = C : , B = C. For, as 4 = 7,

B mu ft = C.

i 7 4

BOOK V. PROP. X. THEOR.

HAT magnitude which has a greater ratio than
another has unto the fame magnitude , is the greater
of the two : and that magnitude to which the fame
has a greater ratio than it has unto another mag¬

nitude , is the lefs of the two.

Letfl : CZ# .* , then CZ • .

For if not, let = or Z] yf \
then, : = % : | (B. 5. pr. 7) or

| Z] £ : i (B. 5. pr. 8) and (invert.),
which is abfurd according to the hypothecs.

/. V') is not = or ZI # , an d
•\ muft be CZ | •

Again, let | :§ C : ; ,

then, • ^ ? ■

For if not, £) muft be C or ~ ,

then : • Zl ■ : \ (B. 5. pr. 8) and (invert.);

or : #= 1 : t!? (B-5-pr.7),whichisabfurd(hyp.);

0 is not C or= ,
and I muft be 33

That magnitude which has, &c.

BOOK F. PROP. XL THEOR.

F 5

ATIOS that are the fame to the fame ratio , are the
fame to each other.

Let ' : == % : and | | : = A : •,

then will > : § j zz A : •.

For if M r . zz, or Z 3 m §J;,
then M || =, or ;□ m ,

and if M HZ, =, or Z 3 ^ >

then M A CZ, zz, or H] m •, (B. 5. def. 5) ;

if M C, =, or 3 m | 9 M A CZ, =, or m «,
and (B. 5. def. 5) : | zz A : •.

/. Ratios that are the fame, &c.

176

BOOK V. PROP. XII. THEOR.

Let

F any number of magnitudes be proportionals , as
one of the antecedents is to its confequent , fo fall
all the antecedents taken together be to all the
confequent s.

then will £ : 0 —

+ 0 + + * + A: # + 0 + + * + •

For if M I CZ m • , then M Q CZ m 0 >,
and M ^ C m M « C m t,
alfo M A I //z o. (B. 5. def. 5.)

Therefore, if M fl| SZ 1 m I) 9 then will

M | -J- M □ + M + M • + M a,

M (E + D + + • + A ) be grater

1 + m 0 - + m W + m ▼ m ©,
or m ( + 0+ + » + •)•

or
than m

In the fame way it may be fhown, if M times one of the
antecedents be equal to or lefs than m times one of the con-
fequents, M times all the antecedents taken together, will
be equal to or lefs than m times all the confequents taken
together. Therefore, by the fifth definition, as one of the
antecedents is to its confequent, fo are all the antecedents
taken together to all the confequents taken together.

If any number of magnitudes, See.

BOOK V. PROP. XIII. THEOR.

1 77

F the firji has to the fecond the fame ratio which
the third has to the fourth, but the third to the
fourth a greater ratio than the fifth has to the
fixth; the firft Jhall alfo have to the fecond a greater
ratio than the fifth to the fixth.

Let : □ = ■ : , but ■ : EZ O : #>

then f : O C : %•

For, becaufe | : C O: •, there are fome mul¬

tiples (M 7 and m) of | and and of and
fuch that M' | EZ m ,

but M' < 2 > not C m 4 $, by the feventh definition.

Let thefe multiples be taken, and take the fame multiples

of qp and Q.

(B. 5. def. 5.) if M' -p [Z, ”, or Z] ni fj ;
then will M' U C, =, or ^ m ,
but M' |Cw' (conftrudlion);

M' [= m Q,

but 1 VT is not Cm' % (conftrudlion) ;
and therefore by the feventh definition,

P:DCC:#.

If the firfl has to the fecond, &c.

17B BOOKV. PROP. XIV. THEOR.

F the firfl has the fame ratio to the fecond which the
third has to the fourth; then, if the firft be greater
than the third , the fecond Jhall be greater than the
fourth ; and if equal , equal; and if lefs , lefs.

Let and firft fuppofe

P C 1 , then will Q C .

For f □ (B. 5 . pr. 8), and by the

hypothefis, ;) \ Q = : > ;

I : > C : □ (B. 5- pr. 13),

Z 1 □ (B. 5. pr. 10.), or 0 C •

Secondly, let “ , then will U —

For ^ = : D (B. 5- pr. 7),

and ^ = B : (hyp.);

:□= : (B. 5. pr. 11),

and □ = (B. 5, pr. 9).

Thirdly, if Z 3 li > then will ^ ' j
becaufe C and ! — 1 t ! C i

1 C U? hy the firft cafe,
that is, U ^

If the firft has the fame ratio, &c.

BOOK V. PROP . XV. THEOR .

l 79

AGNITUDES have the fame ratio to one another
which their equimultiples have .

Let £ and | be two magnitudes;
then, £ : | :: M' : M

For

•• 4 ® • 4 • (B* 5 * P 1 * I2 )-

And as the fame reafoning is generally applicable, we have

:: M' • : M' .

Magnitudes have the fame ratio, &c.

BOOK F. DEFINITION XIII.

180

DEFINITION XIII.

The technical term permutando, or alternando, by permu¬
tation or alternately, is ufed when there are four propor¬
tionals, and it is inferred that the firft has the fame ratio to
the third which the fecond has to the fourth; or that the
firft is to the third as the fecond is to the fourth: as is
fhown in the following propofition :—

Let : + : g,

by “ permutando” or “alternando” it is
inferred ; ^ ^ : ;g .

It may be neceftary here to remark that the magnitudes
■ ? $ tp, , mud; be homogeneous, that is, of the

fame nature or fimilitude of kind; we muft therefore, in
fuch cafes, compare lines with lines, furfaces with furfaces,
folids with folids, &c. Hence the ftudent will readily
perceive that a line and a furface, a furface and a folid, or
other heterogenous magnitudes, can never Hand in the re¬
lation of antecedent and confequent.

BOOK V. PROP. XVL THEOR.

181

F four magnitudes of the fame kind be proportionals ,
they are alfo proportionals when taken alternately.

Let qp : □ :: : # , then : :: O :

ForMf :MQ::f :□ (B. 5. pr. 15),
and M fp : M U :: I : (hyp.) and (B. 5. pr. 11);

alfo m : m ♦ : 4 (B. 5. pr. 15);

M p : M Q :: ^ : m (B. 5. pr. 14),

and if M ^ C, =, or Z 3 ™ 3 ,
then will M ^ or I m (B. 5. pr. 14);

therefore, by the fifth definition,

•% If four magnitudes of the fame kind, &c.

182

BOOK V. DEFINITION XVI

DEFINITION XVI.

Dividendo, by divifion, when there are four proportionals,
and it is inferred, that the excefs of the firft above the fecond
is to the fecond, as the excefs of the third above the fourth,
is to the fourth.

Let A : B :: C : ;

by “ dividendo” it is inferred
A minus B : B : :"C minus : ■.

According to the above, A is fuppofed to be greater than
B, and C greater than ; if this be not the cafe, but to
have B greater than A, and greater than Q, B and D
can be made to Hand as antecedents, and A and C as
confequents, by “ invertion ”

B : A : •. D : C ;

then, by “dividendo,” we infer
B minus A : A :: minus :C .

BOOK V. PROP. XVII. PHEOR.

F magnitudes , taken jointly , be proportionals , they
fhall alfo be proportionals when taken feparately :
that is, if two magnitudes together have to one of
them the fame ratio which two others have to one
of thefe, the remaining one of the firfi two Jhall have to the other
the fame ratio which the remaining one of the loft two has to the
other of thefe.

Let W + O : O :: + v ,

then will V • O :: H • •

Take M ^ Cl m □ to each add M u,
then we have M w + m u nz U + m D,
orM(f + O) C (« + M) □:
but becaufe + Q : Q :: I + : ♦ (hyp.),

andM(* + 0)C(« + M) Q;

M ( , + r ’) C (» + M) 4 (B. 5. def. 5);

M l + M ♦ C m + M ;

M M l m ^ 9 by taking M \ - from both Tides :
that is, when M 1 m O, then M C m ♦ .

In the fame manner it may be proved, that if
M ^ zz or | m then will M zz or —I m ^ :
and V ♦ (B. 5. def. 5 ).

If magnitudes taken jointly, &c.

I

BOOK V. DEFINITION XV.

DEFINITION XV.

The term componendo, by compofttion, is ufed when there
are four proportionals ; and it is inferred that the firft toge¬
ther with the fecond is to the fecond as the third together
with the fourth is to the fourth.

Let A : B :: : D ;

then, by the term “ componendo,” it is inferred that
A + B : B :: + D : D.

By 44 invertion” B and D may become the firfl and third,
A and C the fecond and fourth, as

B : A :: J) : ,

then, by “ componendo,” we infer that

B —J— A : A :; —j—

BOOK V. PROP. XVIII. THEOR.

F magnitudes , taken feparately , be proportionals,
they fhall alfo be proportionals when taken jointly:
that is , if the jirjl be to the fecond as the third is
to the fourth , the Jirft and fecond together fhall be
to the fecond as the third and fourth together is to the fourth.

Let qp : Q :: ' ,

then ^ + 0:0:: + : > ;

for if not, let ^ -j" O : O :: + ® :

fuppoling not ~ ;

V '■ □ :: : • ( B - s- P r -17);

but W : U :: : (hyp.);

: £ :: : (B. 5. pr. 11);

O = (B. 5. pr. 9),

which is contrary to the fuppolition;

@ is not unequal to ^ \
that is 0 zz ;

W + O : O :: + • *

•\ If magnitudes, taken feparately, &c.

B B

186

BOOK F. PROP. XIX. THEOR.

F a whole magnitude be to a whole , as a magnitude
taken from the firfl , is to a magnitude taken from
the other; the remainder fhall be to the remainder ,
as the whole to the whole.

Let ^ + □ : | + :: p :

then will Q : :: • ' -j- O : 1 + ► >

For P + Q : qp :: p + > : ■ (alter.),

O : :: ♦ : fl (divid.),

again □ : :: W '■ ■ (alter.),

but W + O '• ■ + ♦ " V '• ■ hyp.);

therefore : :: V “I" O • -j-

(B. 5. pr. 11).

If a whole magnitude be to a whole, &c.

DEFINITION XVII.

The term “ convertendo,” by converfion, is made ufe of
by geometricians, when there are four proportionals, and
it is inferred, that the firft is to its excefs above the fecond,
as the third is to its excefs above the fourth. See the fol¬
lowing proportion :—-

BOOK V. PROP. E. THEOR.

187

F four magnitudes be proportionals , they are alfo
proportionals by converfon: that is, the firf is to
its excefs above the fecond, as the third to its ex -
cefs above the fourth.

Let §0 :<>::■ : ♦»

then fhall 11 O : © :: H 4 : I >

Becaufe • O :

therefore | : O S • ^ (divid.),

O : # :: $ : B (inver.),

# O : B :: ■ : B (compo.)-

If four magnitudes, &c.

DEFINITION XVIII.

“ Ex $quali” (fc. diflantia), or ex asquo, from equality of
diflance : when there is any number of magnitudes more
than two, and as many others, fuch that they are propor¬
tionals when taken two and two of each rank, and it is
inferred that the firfl is to the lafl of the firfl rank of mag¬
nitudes, as the firfl is to the lafl of the others: “ of this
there are the two following kinds, which arife from the
different order in which the magnitudes are taken, two
and two.”

188

BOOK V. DEFINITION XIX.

DEFINITION XIX.

“ Ex aequali,” from equality. This term is ufed limply by
itfelf, when the firft magnitude is to the fecond of the firft
rank, as the firft to the fecond of the other rank; and as
the fecond is to the third of the firft rank, fo is the fecond
to the third of the other; and fo on in order: and the in¬
ference is as mentioned in the preceding definition; whence
this is called ordinate proportion. It is demonftrated in
Book 5. pr. 22.

Thus, if there be two ranks of magnitudes,
A , B , , , , F, the firft rank,

and L, M, , , P, Q, the fecond.

fuch that A : B :: L : M , B : :: M : ,

: :: : , : :: : , : :: : ;

we infer by the term “ ex sequali” that

A : F :: L : Q.

BOOK V. DEFINITION XX.

189

DEFINITION XX.

“ Ex sequali in proportione perturbata feu inordinata,”
from equality in perturbate, or diforderly proportion. This
term is ufed when the firft magnitude is to the fecond of
the firft rank as the laft but one is to the laft of the fecond
rank; and as the fecond is to the third of the firft rank, fo
is the laft but two to the laft but one of the fecond rank;
and as the third is to the fourth of the firft rank, fo is the
third from the laft to the laft but two of the fecond rank;
and fo on in a crofs order: and the inference is in the 18th
definition. It is demonftrated in B. 5. pr. 23.

Thus, if there be two ranks of magnitudes,

A , B , , D , , , the firft rank,

and , , N , O , P , Q , the fecond,

fuch that A : B :: P : Q , B : C :: O : P ,

C : D :: : O, D : :: : N, : :: : ;

the term “ ex aequali in proportione perturbata feu inordi¬
nata ” infers that
A : F :: L : Q .

190

BOOK V. PROP. XX. THEOR.

F there be three magnitudes , and other three , which ,
taken two and two , have the fame ratio ; then , z/'
the firft be greater than the third , the fourth fhall
be greater than the fixth ; and if equal , equal;

and if lefs> lefs.

Let 9 , 0 , 9 , be the firft three magnitudes,
and ♦ , 0 , 9 , be the other three,
fuch that 9 : U " ♦ O , and O :■ ::£>:••

Then, if C, =, or I] , then will C, =,

or Zl #•

From the hypothefis, by alternando, we have

9 :♦ " 0 : 0 ,

and O : O :: ■ : 5

9 :♦ " (B. j. P r. n);

/. if d, =, or Z1 , then will C, =,
orZ 3 ( B - 5 - P r - H)-

If there be three magnitudes, &c.

BOOK V. PROP. XXL THEOR.

191

F there be three magnitudes , and other three which
have the fame ratio, taken two and two , but in a
crofs order; then if the firfi magnitude be greater
than the third , the fourth fhall be greater than the
fixth ; and if equal, equal; and if left , lefs.

Let ’ , £ 9 5 be the firft three magnitudes,

and •• the other three,

fuch that ^ : £ :: <2> and ^ : O •

Then, if d, =, or 33 | , then
will ♦ c, □ ».

Firft, let be £2 | :
then, becaufe £ is any other magnitude,

•‘A c ■ : A ( B - 5- P r - 8 );

hut : :: : A (hyp-);

0> : C : A ( B - 5- P r - r 3);

and becaufe A =■ "♦ : O ( h yp-);

• Hi : A "O : V (inv.),
and it was fhown that > : C B • A: ?

• •• C •' a O '■ C (B. j. pr. 13);

192

BOOK V. PROP. XXL THEOR.

=1

that is C

Secondly, let zz j 5 then fhall zz

For becaufe zz § ,

: A = . : A (B. 5. pr. 7);

but p : — <2> : (hyp.),

and : 4 ; ~ 0> : (hyp- and inv.),

O : = O : (B. 5. pr. 11),

= ( B - 5 - P r - 9 )-

Next, let be Z] , then fhall be Z 2

for ■ n. v,

and it has been (hown that : A = O : # ,
and A : = : O ;

by the firft cafe is C >
that is, Z3

If there be three, &c.

BOOK V. PROP. XXII. THEOR.

I 93

F there be any number of magnitudes , and as many
others , which, taken two and two in order, have
the fame ratio ; the firjl fhall have to the laft of
the firji magnitudes the fame ratio which the firjl
of the others has to the lafi of the j'ame.

N.B.— ■ This is ufually cited by the words “ ex aqualif or
“ex cequo

Firfl, let there be magnitudes ^ , | 9
and as many others ^ , <f), ,

fuch that

* :♦ ”♦ : C>

and 4 ^ : | ::<> : r 5

then fhall p : r : 4 •

Let thefe magnitudes, as well as any equimultiples
whatever of the antecedents and confequents of the ratios,
fland as follows :—

V>+> -fO. ,

and

becaufe ^ : < > ;

.% M fp : m :: M ^ : m (B. 5 . p. 4 ).

For the fame reafon
m + :N| :: m 0 : N ;
and becaufe there are three magnitudes.

194

BOOK V. PROP. XXII. THEOR.

-M- 'V 5 171 ">■ 5 N ,
and other three, M ^ , m o> N >
which, taken two and two, have the fame ratio;

ifMfP C=,=, orZ]N

then will M t => or ^ N , by (B. 5. pr. 20)
and /. f : | :: ♦ : (def. 5).

Next, let there be four magnitudes, W>
and other four,

which, taken two and two, have the fame ratio,
that is to fay, ^ • O : ® >

and 1 : : I : ▲ ?

then fhall : ^ :: ^ ;

for, becaufe ? ? are three magnitudes,

and 3 , , other three,

which, taken two and two, have the fame ratio;
therefore, by the foregoing cafe, : 0 ::0 : ,

but : ■ :: : A. ;

therefore again, by the firil call-, : : A. ;

and fo on, whatever the number of magnitudes be.

/. If there be any number, &c.

BOOK V. PROP. XXIII. PHEOR.

J 95

F there be any number of magnitudes , and as many
others , which , taken two and two in a crofs order,
have the fame ratio ; the firf fhall have to the laft
of the firft magnitudes the fame ratio which the

firf of the others has to the lafl of the fame.

N.B .—This is ufually cited by the words
proportione perturbataor “ ex aequo perturbato

“ ex aequail in

Firft, let there be three magnitudes, H 9

and other three, , fy, £ ,
which, taken two and two in a crofs order,
have the fame ratio;
that is, : 0 :: (> : £ ,
andQ :: 4 : 0 ,

then fhall : | : : '| .

Let thefe magnitudes and their refpective equimultiples
be arranged as follows:—

^ U > H > > O > • 9

then :Q :: M :MQ (B. 5. pr. 15);
and for the fame reafon

O : • :: m o : m • ;

but ■ :q ::< 3 (hyp.),

196 BOOK V. PROP. XXIII. THEOR.

M : M C '1 :: O : # (B. 5. pr. 11);
and becaufe D ' ■ :: : o (%P-)>

M ■ ’ I rn | :: : m Q. (B. 5. pr. 4);

then, becaufe there are three magnitudes,

■-

and other three, M , m O, m £.
which, taken two and two in a crofs order, have

the fame ratio;

therefore, if M C, => or m § 1 ?
then will M C, z=, or ^ m £ (B. 5. pr. 21),
and /, : | : $ (B. 5. def. 5).

Next, let there be four magnitudes,

? O 5 B? ^

and other four, m, m, A,

which, when taken two and two in a crofs order, have

the fame ratio ; namely,

V :D ::

o-m ■■

and ■ : :: O

then fhall I " O

For, becaufe , B are three magnitudes,

BOOK V. PROP. XXIII. THEOR.

197

and i | 5 m 5 Jk 9 other three,

which, taken two and two in a crofs order, have

the fame ratio,

therefore, by the firft cafe, : Sf :: ! A .

but ■ : :: O : ’

therefore again, by the fird: cafe, : :: • A. j

and fo on, whatever be the number of fuch magnitudes.

If there be any number, &c.

198 BOOK V. PROP. XXIV. THEOR.

F the firfi has to the fecond the fame ratio which
the third has to the fourth , and the fifth to the
fecond the fame which the fixth has to the fourth ,
the firfi and fifth together Jhall have to the fecond
the fame ratio which the third and fixth together have to the
fourth.

First.

Second.

Third.

Fourth,

V

□

■

♦

Fifth.

Sixth.

O

•

Letjp ,

and £> ; Q :; © : ,

then +

For <2> : □ :: : (hyp.),

and (hyp.) and (invert.),

0 : W :: ■ : ® ( B - 5 -P r - 22 )>

and, becaufe thefe magnitudes are proportionals, they are
proportionals when taken jointly,

* + O : O :: + : ( B - 5- P r - i8 )>

hut O : D :: • : ( h yp-)>

.*. W + ' ! o •• “h • (F. 5- P r * 22 ) -

If the firft, &c.

BOOK V. PROP . XXV. THEOR.

199

F four magnitudes of the fame kind are propor¬
tionals, the greateft and leaf of them together are
greater than the other two together .

Let four magnitudes, g| -|- } and ,

of the fame kind, be proportionals, that is to fay,

f + □ : ■ + :: D : >

and let + □ be the greateft of the four, and confe-
quently by pr. A and 14 of Book 5, is the leaft;

then will $ + Q + be C + + O j

becaufe *+□:* + :: O : <>,

W : ■ :: W + O : B + ( B - 5 - P r -19)-

kt f +Dc + (hyp.),

w C (B. 5. pr. A) ;
to each of thefe add □ + ,

•••*+□ + !=■ + □+ .

If four magnitudes, &c.

200

B00KV. DEFINITION X.

DEFINITION X.

When three magnitudes are proportionals, the firit is faid
to have to the third the duplicate ratio of that which it has
to the fecond.

For example, if A, , C , be continued proportionals,
that is, A : :: : C , A is faid to have to C the dupli¬
cate ratio of A : ;

A

or — zz the fquare of —.

This property will be more readily feen of the quantities
ar * 9 , a > for a r* : ■ :: : a ;

and — zr J z; the fquare of — zz r.

or of a.

, a r~ ,

for — sz zz the fquare of — zz —

a r * r 2 ^ r

DEFINITION XI.

When four magnitudes are continual proportionals, the
firft is faid to have to the fourth the triplicate ratio of that
which it has to the fecond; and fo on, quadruplicate, &c.
increaling the denomination Hill by unity, in any number
of proportionals.

For example, let A, , C, D, be four continued propor¬
tionals, that is, A : :: : :: : D ; is faid to have

to D, the triplicate ratio of A to ;

A _i r A

BOOKF. DEFINITION XL

201

This definition will be better underflood, and applied to
a greater number of magnitudes than four that are con¬
tinued proportionals, as follows :—

Let ar s , , • r , a, be four magnitudes in continued pro¬
portion, that is, a r s : :: ■' ’ar • •ar '-a,

S 5

then — z: r 3 ~ the cube of 1 — zz r.
a

Or, let ar 5 , ar 4 , ar 3 , ar 2 , ar, a, be fix magnitudes in pro¬
portion, that is

ar b : ar 4 :: ar 4 * ar 3 :: ar 3 : aF :: ar° : ar :: ar : a,

ci r * 5 - a y*'*

then the ratio — zz F zn the fifth power of — 2 = r.

a r ar 4

Or, let a, ar, ar 2 , ar 3 , ar 4 , be five magnitudes in continued

proportion; then = ~ z= the fourth power of — =

ar 4 F r ar r

DEFINITION A.

To know a compound ratio :—

When there are any number of magnitudes of the fame
kind, the firfi: is faid to have to the laffc of them the ratio
compounded of the ratio which the firfi: has to the fecond,
and of the ratio which the fecond has to the third, and of
the ratio which the third has to the fourth ; and fo on, unto
the lafb magnitude.

For example, if A, B, C, D,
be four magnitudes of the fame
kind, the firfi A is faid to have to
the lafl ! ) the ratio compounded
of the ratio of A to B, and of the
ratio of B to C, and of the ratio ofC to D ; or, the ratio of

A B C D

E F G H K L

M N

D D

202

BOOK V. DEFINITION A.

A to P is faid to be compounded of the ratios of A to ,
B to C, and C to D .

And if A has to B the fame ratio which has to 1 , and
B to C the fame ratio that (.< has to H, and ( to D the
fame that K has to ; then by this definition, A is said to
have to D the ratio compounded of ratios which are the
fame with the ratios of E to F, to H, and K to L. And
the fame thing is to be underflood when it is more briefly
expreffed by faying, A has to D the ratio compounded of
the ratios of E to E, G to H, and K to s .

In like manner, the fame things being fuppofed; if
has to the fame ratio which A has to D , then for fhort-
nefs fake, is faid to have to the ratio compounded of
the ratios of R to F, G to , and K to L.

This definition may be better underflood from an arith¬
metical or algebraical illufixation ; for, in fact, a ratio com¬
pounded of feveral other ratios, is nothing more than a
ratio which has for its antecedent the continued produdt of
all the antecedents of the ratios compounded, and for its
confequent the continued product of all the confequents of
the ratios compounded.

Thus, the ratio compounded of the ratios of
2:3, : , 6 : 1 j 5 2 : 5,

is the ratio of 2 X X 6 X 2 : 3 X X1X5,
or the ratio of 96 : 1155, or 32 : 385.

And of the magnitudes A, B, , , h, P, of the fame

kind, A : F is the ratio compounded of the ratios of
A : B, B : C, : , : E, E : F;

for A X B X X

or

A X B X X

B X C X X E X F

D X E : R x C X D x E x F,

X , or the ratio of A : F.

BOOK V. PROP. F. THEOR.

203

ATI OS which are
are the fame to one

compounded of the fame ratios
another.

Let A : : F : ,

B : C :: G : H,
C : D :: H : K,
and : E • • : L.

ABODE
F G H K L

Then the ratio which is compounded of the ratios of
: , : , : D, : E, or the ratio of A : E, is the

fame as the ratio compounded of the ratios of P : ,

: , : , : L , or the ratio of F : L.

D

K

and

. AX X X _ F X G X H X

* * X X X E X X XL’

or the ratio of A : E is the fame as the ratio of :

L.

The fame may be demonfhrated of any number of ratios
fo circumflanced.

Next, let A : :: : L,

B : C :: H : K,
C : D :: G : H,
: E :: F : G s

204

BOOK V . PROP . E. THEOR.

Then the ratio which is compounded of the ratios of
A : , : , : , : E, or the ratio of A : E, is the
fame as the ratio compounded of the ratios of : , : ,

: , F : , or the ratio of :L.

F

vjr

3

e

• 9

AX X X D

X X

and

X E

.

’ • E

X X X

L X X X ’

L ’

or the ratio of A : E is the fame as the ratio of : L.

•% Ratios which are compounded, &c.

BOOK F. PROP . G. THEOR.

20 5

F fever a l ratios be the fame to fever al ratios, each
to each , the ratio which is compounded of ratios
which are the fame to the firfl ratios , each to each ,
Jhall be the fame to the ratio compounded of ratios
which are the fame to the other ratios , each to each.

ABCDEFGH PQRST
a bed e f g h

If A : B

C: D
E : F

and G : H

and : :: P : Q a : b

: D :: Q : R c : d
h: F :: R:S e :f

g:h

G:H::S:T \ g : A

then P : T n;

2 — £ —

R D —”

R _ K _

S F

± _ _G _

T H

onrl • ' XgXRXS X X X

• • Q x R X S X T — x X X~ *

and

or P : T — : .

• •

If feveral ratios, &c.

206

BOOK V. PROP. H. THEOR.

F a ratio which is compounded of feveral ratios be
the fame to a ratio which is compounded of feveral
other ratios ; and if one of the firfi ratios , or the
ratio which is compounded of feveral of them , be
the fame to one of the laft ratios , or to the ratio which is com¬
pounded offeveral of them; then the remaining ratio of the firfi ,
or , if there be more than one> the ratio compounded of the re¬
maining ratios , Jhall be the fame to the remaining ratio of the
lafl , or , if there be more than one , to the ratio compounded of thefe
remaining ratios.

ABCDEFGH

PQRSTX

Let A : B, B : C, C : D, D : E, E : F, F : G, G : H,
be the firfi: ratios, and P : Q, Qj G R : S , S : f, : X ,

the other ratios; alfo, let A : H, which is compounded of
the firfi: ratios, be the fame as the ratio of : , which is

the ratio compounded of the other ratios; and, let the
ratio of A : E, which is compounded of the ratios of A : B,
B : C, C : D, D : E, be the fame as the ratio of P : K ,
which is compounded of the ratios : Q^Qj G

Then the ratio which is compounded of the remaining
firfi: ratios, that is, the ratio compounded of the ratios
E : F, F : G, G : H, that is, the ratio of E : H, fhall be
the fame as the ratio of B : X, which is compounded of
the ratios of R : 3, S : T, : X, the remaining other

ratios.

BOOK F. PROP. H. PHEOR.

207

r> erfl11 f p AXBXCXDXEXPXG „ F X O X
BXCXDXEXFXGXH“SXKX

A X B X C X 1) w E X F X G _ X 9 y

BXcXdXE ^ FXGXH O X H A

and A X B X C X D _ r> X O
BXCXDXE fiX K*

• E x F X G «-», R X s X r
* * F x G X H s X ! X x*

• E —

IF

E •' H

R

X.

*% If a ratio which, &c.

R X S X T

x r x X ?

X X T

S' X T x x 5

2o8

BOOK V ; PROP. K. THEOR.

F there be any number of ratios, and any number of
other ratios, fuch that the ratio which is com¬
pounded of ratios, which are the fame to the firft
ratios, each to each, is the fame to the ratio which
is compounded of ratios, which are the fame, each to each, to
the loft ratios—and if one of the firf ratios, or the ratio which
is compounded of ratios, which are the fame to feveral of the
firf ratios, each to each, be the fame to one of the lafi ratios,
or to the ratio which is compounded of ratios, which are the
fame, each to each, to feveral of the lafi ratios—then the re¬
maining ratio of the firft; or, if there be more than one, the
ratio which is compounded of ratios, which are the fame, each
to each, to the remaining ratios of the firft, fhall be the fame
to the remaining ratio of the lafi, or, if there be more than
one, to the ratio which is compounded of ratios, which are the
fame, each to each, to thefe remaining ratios.

h k m n s
AB, CD, EF, GH, K L, MN,

J , O K ,

abed

W, X Y,
e f g

a b c d e f g

h k l m n p

Let A : B, C :D, E :F, G :H, K :L, M :N, be the
firft ratios, and : , : , : , : , : , the
other ratios;

and let A : B zz a : ,

C : D zz b :c ,

E : F zz c :d,

G : H zz d : e,

K : L =z r :/,

M : N zz :g.

BOOK V. PROP. K. THEOR. 209

Then, by the definition of a compound ratio, the ratio
of ^ is compounded of the ratios of a m .c»c '-did

e : y, f * or, which are the fame as the ratio of A : B, C : D>
E : F, G : H, K : L, M : N, each to each.

Alfo, : zz 2 h \ k,

: =: k : /,

: zzz l: m,

V : V zz m : n,

: zz n \ p.

Then will the ratio of h:p be the ratio compounded of
the ratios of h : k, k : /, /: m, m:n } n:p , which are the
fame as the ratios of : , : , : , : , : ,
each to each.

/. by the hypothefis a : g zz h:p.

Alfo, let the ratio which is compounded of the ratios of
A: B, C : D, two of the firil ratios (or the ratios of a : c,
for \ : B zz a : and C : D zz £: c ), be the fame as the
ratio of a: d, which is compounded of the ratios of a : b,
b : c, c : d, which are the fame as the ratios of : ,

: , : , three of the other ratios.

And let the ratios of h : s, which is compounded of the
ratios of h : k, k : m, m : n, n : s, which are the fame as
the remaining firil ratios, namely, E : F, G : H, K : L,
M : N ; alfo, let the ratio of e : g, be that which is com¬
pounded of the ratios e : f, f: g, which are the fame, each
to each, to the remaining other ratios, namely, : ,

: . Then the ratio of h ; s lhall be the fame as the

ratio of e : g; or h : s zz e : g.

PW A X C X E X G X K x M __ fl X iX cX jf'X *X /
BXDXFXHXLXN ~ b X „ X d X « X /X g’

E E

210

BOOK V. PROP. K. THEOR.

1 X X X X _ hx k X l XmXn

X X X X kXlXmXnXp ’

by the compofition of the ratios ;

. aXbXcXdXeXf _ hXkX l XmXn /j n

* * bX cX dX eXfX g kXlXmXnXp {

or a ^ ^ NX c X d X e X f _ h x k X l vy m X n

b X c ^ d X e X f X g k X l X m ^ n X p ’

a X b _ A X C _ X X _ a X b X c _ h Xk X /

IX C -B XB X X b X c X d k X l X rn

• c X d X f X f _ m X n

’ • dXeX/Xg n Xp'

A „ j cXdXeXf

And ix7xf>r R

h X k X

k X m X n X s

(hyp.),

and

m X n _ e X f

n X P

f X g

(hyp.).

• b X k X m X n __ e_f

•• k X m X n X s f g ’

B ° 0 If there be any number, &c.

*, * Algebraical and Arithmetical expositions of the Fifth Book of Euclid are given in
Byrne's Doctrine of Proportion ; published by Williams and Co. London. 1841.

BOOK VI.

DEFINITIONS.

L

ECTILINE AR
figures are faid to
be fimilar, when
they have their fe-
veral angles equal, each to each,
and the lides about the equal
angles proportional.

II.

Two Tides of one figure are faid to be reciprocally propor¬
tional to two fides of another figure when one of the fides
of the firfl is to the fecond, as the remaining fide of the
fecond is to the remaining fide of the firft.

III.

A straight line is faid to be cut in extreme and mean
ratio, when the whole is to the greater fegment, as the
greater fegment is to the lefs.

IV.

The altitude of any figure is the straight line drawn from
its vertex perpendicular to its bafe, or the bafe produced.

2 )2

BOOK VI. PROP. I. THEOR ,

Let the triangles J| and
have a common vertex, and
their bafes and

RIANGLES

and parallelo¬
grams having the
fame altitude are
to one another as their bafes.

in the fame lfraight line.

Produce ■■ ■ both ways, take fucceffively on

produced lines equal to it; and on ... pro¬

duced lines succeffively equal to it; and draw lines from
the common vertex to their extremities.

The triangles

thus formed are all equal

to one another, lince their bafes are equal. (B. i. pr. 38.)

and its bafe are refpectively equi¬

multiples of HI and the bafe

BOOK VI. PROP. I. THEOR.

2- 3

In like manner

and its bafe are refpec-

tively equimultiples of and the bafc

• •

If m or 6 times
then m or 6 times

C = or ZD 71 or 5 times
C = or ^ n or 5 times hmw 9
m and n fland for every multiple taken as in the fifth
definition of the Fifth Book. Although we have only
fhown that this property exifts when m equal 6, and n
equal 5, yet it is evident that the property holds good for
every multiple value that may be given to m , and to n.

(B. 5. def. 5.)

Parallelograms having the fame altitude are the doubles
of the triangles, on their bafes, and are proportional to
them (Part 1), and hence their doubles, the parallelograms,
are as their bafes. (B. 5. pr. 15.)

Q. E. D.

214 BOOK VI. PROP. II. THEOR.

a Jlraight line ■» ■■■ »

he drawn parallel to any

<ide ■■■■■■■■■■» of a tri¬
angle, it fhall cut the other
lides, or thofe fdes produced, into pro¬
portional fegments.

And if any ftraight line ——.

divide the fdes of a triangle, or thofe
fides produced, into proportional feg¬
ments, it is parallel to the remaining

fde •■■■■■■■■■■».

PART I.

Let ■■■■■■■■ || , then fhall

' * mmrnmnummm J J ■■ * OS HUM KM ■!!> .

Draw

and

(B. i. pr. 37);

(B.5-pr.7); but

(B. 6. pr. i),

• *

• •
• •

J itittliltsiat #

(B. 5. pr. n).

BOOK VL PROP. II. THEOR.

21 5

PART II.

Let

then

• e
9 •

II

Let the fame conftrudtion remain,

becaufe

and

hut

}> (B. 6. pr. i);

St

* \

" V

\ : i \ (B.5. pr. II.)

(B. 5. pr. 9);

A

but they are on the fame bafe -■■■■«**«■ and at the

fame fide of it, and

•°* - II .. (B. 1. pr. 39).

O. E. D.

2l6

BOOK VI. PROP. III. THEOR.

RIGHT line (

- “)

bifedting the angle of a

triangle , divides the op-

pofite fide into fieg?nents

proportional

to the conterminous fides L

- )•

And if a firaight line (

— )

drawn from any angle of a triangle

divide the oppofite fide (■■■.——)

into fegments (——— ? ■■■ ■■■...dj

proportional to the conterminous fides ( ■■■■. 9 _),

it bifedls the angle.

PART I.

Draw || ... — 9 to meet *

then, 4=4 (B. i. pr. 29),

•• = 4 ; but ^ = 4 -,

mum m mmmm «■

and becaufe

(B. 1. pr. 6);

ll

mm utm

• •
• •

(B. 6. pr. 2);
but -- m-mmu-mm*mm - -----

■- :

(B. 5. pr. 7).

BOOK KI. PROP. III. THEOR.

217

and

but

• •

PART II.

Let the fame confirmation remain,

(B. 6. pr. 2);

WMUUUMm

( h yp-)

iiiiaaoi

(B. 5. pr. 11).

(B. 5. pr. 9),

and /.

and 4=1 (B. 1. pr. 5); but fince

.. II *.— = t.

and =

(B. 1. pr. 29);

4 zz ; ' 5 and zz

and /. —■«, ■« bifedls *

E. D.

F F

2 I 8

BOOK VI. PROP. IV. THEOR.

N equiangular tri¬
angles (

and

\) the files
about the equal angles are pro¬
portional, and the Jides which are
oppojite to the equal angles are
homologous.

Let the equiangular triangles be fo placed that two lides
—» oppofite to equal angles

and

<2 may be conterminous, and in the fame ftraight line;
and that the triangles lying at the fame lide of that ftraight
line, may have the equal angles not conterminous,

, and A.* A.

l. e.

oppofite to

Draw and

Then, becaufe

AA

, . . . || —«*»»»■* (B.I.pr.28);

and for a like reafon, **.—■—* ||

. / . 7

is a parallelogram.

But

(B. 6 . pr. 2);

BOOK FI. PROP . IV. THEOR.

219

and lince

• o

• •

(B. 1. pr. 34),

• and by

alternation.

(B. 5. pr. 16).

In like manner it may be fhown, that

® mm a it • • i« a* ••

e • •

0 IllllllllllUi ’

and by alternation, that

0 0 •«***)■:•» • a mmmtaamnmn 0

but it has been already proved that

* * ■■■■■■aBSSB * •■■»■■«■■■■«

e • 0

and therefore, ex asquali,

• • *m.m .m -9'MWMma • i«n

• • •

(B. 5. pr. 22),

therefore the lides about the equal angles are proportional,
and thofe which are oppofite to the equal angles

are homologous.

Q. E. D.

220

BOOK VI. PROP . V. THEOR .

F two triangles have their Jides propor¬
tional ( * ■■■!<■»■■■

wmmHmmmm • ) and

• •
• •

(

« •
• •

A

. I " ) they are equiangular ,

tfW the equal angles are fubtended by the homolo¬
gous Jides .

\

From the extremities of

draw

A..4

and

• •••« ■ i

making

(B. i. pr. 23);

and confequently r= (B. 1. pr. 32),

and fince the triangles are equiangular.

aimillia * *

but

(B. 6. pr. 4) ;

immm * • -MMMM

(hyp.);

and confequently

9 •
• 9

(B. 5. pr. 9).

In the like manner it may be fhown that

BOOK VI. PROP . V ; THEOR.

22 I

Therefore, the two triangles having a common bafe
i 9 and their lides equal, have alfo equal angles op-

polite to equal lides, i. e.

= w = " ■

(B. i. pr. 8).

But

w = a k

(conlt.)

and A = A \ for the fame

„ \, and

reafon

A

confequently Up — (B. i. 32);

and therefore the triangles are equiangular, and it is evi¬
dent that the homologous lides fubtend the equal angles.

E. D.

222

BOOK VI. PROP. VI. THEOR.

F two triangles ( V

.A

.* »

■ ■•Mai

and

) have one

4 angle ( A

) of the one> equal to one

&

\ angle ( f X ) of the other , and the fides
^ about the equal angles proportional , the
triangles Jhall be equiangular , have

thofe angles equal which the homologous
Jides fubtend.

From the extremities of

one of the lides

of

} about Cs 9 draw
and 9 making

f=4,»i^= A-, 4

(B. i. pr. 32), and two triangles being equiangular.

• * ■■Haifllllli * ®

(B. 6. pr. 4);

but •««*■»■•*•*•»» J ■■■■■■■■■■■ mmtmmammm

( h yp-);

9

9 9

rnmmmavfmAMmm

(B. 5. pr. 11),

and confequently

(B. 5. pr. 9);

BOOK VI. PROP. VI THEOR.

223

/ in every refpedt,
(B. 1. pr. 4).

and

(conft.),

; and

/\ A

lince alio 1 \ in M 9

= (B. 1. pr. 32);

A

and /. *•;.and

A

are equiangular, with

their equal angles oppofite to homologous fides.

Qi E. D.

224

BOOK VI. PROP. VII. THEOR.

F two triangles (

A

/ \

A.

\ ) have one angle in

each equal ( 1 equal to 4 ), the

Jides about two other angles proportional
( — 11 l ~ ll ) ?

and each of the remaining angles (

and ) either lefs or not lefs than a

right angle , the triangles are equiangular , and thofe angles
are equal about which the fdes are proportional.

Firft let it be affumed that the angles and

are each lefs than a right angle: then if it be fuppofed

that and contained by the proportional tides,

are not equal, let 4 be the greater, and make

a

Becaufe

4

zz ' ' (hyp.), and

(conti.)

• •

= 4

(B. i. pr. 32);

BOOK VI. PROP. VII. THEOR.

225

o 0

• ■kavinm

(B. 6. pr. 4),

but

4 Q

• ® •

(hyp-)

• «

and

4,4

(B. 5. pr. 9),
(B. 1. pr. 5).

4

But ^^1 is lefs than a right angle (hyp.)

is lefs than a right angle; and muft

be greater than a right angle (B. 1. pr. 13), but it has been

proved = and therefore lefs than a right angle,

which is abfurd. and are not unequal;

they are equal, and fince 4 = A

( h yp-)

4 = 4 (B. 1. pr. 32), and therefore the tri¬
angles are equiangular.

4 and

But if and be alfumed to be each not lefs

than a right angle, it may be proved as before, that the
triangles are equiangular, and have the fides about the
equal angles proportional. (B. 6. pr. 4).

Q. E. D.

G G

226

BOOK VI. PROP. VIII. THEOR.

N a right angled
triangle

),if

)

a perpendicular (
be drawn from the right angle
to the oppofitefide , the triangles

) on each fide of it are fimilar to the whole

triangle and to each other.

Becaufe

(B. i. ax. 11), and

A = 4

(B. i. pr. 32);

and are equiangular; and

confequently have their iides about the equal angles pro¬
portional (B. 6. pr. 4), and are therefore limilar (B. 6.
def. 1).

In like manner it may be proved that

is limilar to

has been Ihewn to be limilar

limilar to the whole and to each other.

Q E. D.

BOOK VI. PROP. IX. PROB.

ROM a given firaight line ( )

to cut off any required part.

From either extremity of the
given line draw making any

angle with ; and produce

m ! ■ ■ ■ till the whole produced line

contains as 0 ft en as

contains the required part.

Draw

, and draw

ii

is the required part of

For lince —

• •
• •

(B. 6. pr. 2), and by compolition (B. 5. pr. 18);

1 mu •

• •
• •

but ——. contains ■' as often

as contains the required part (conft.);

is the required part.

• •

Q. E. D.

228

BOOK VI. PROP. X. PROB.

draw

Since j

(

O divide a Jlraight
line (— — — )
Jimilarly to a
given divided line

— )•

From either extremity of

the given line - .■ ■ -

draw

making any angle ; take

and

m m ■ m m m

equal to

refpedtively (B. i. pr. 2);
, and draw and

— || to it.

} are II>

• •

© ©

(B. 6. pr. 2),

or

(conft.)j

and

(B. 6. pr. 2),

• ©

0 ©

(conft.).

and the given line
limilarly to

is divided

Q. E. D.

BOOK VI. PROP. XI. PROP.

229

O find a third proportional
to two given firaight lines
( . and ' »"» ).

At either extremity of the given
line ' ■ . draw »

making an angle ; take

and

draw

9

make
and draw

------ II

(B. 1. pr. 31.)

is the third proportional

to ir»rn ■■ ■ ■ . — and .

For fince

II —

tmm ®

(B. 6 pr. 2);

blit «■■■■■■«■■

(conft,);

( B - 5 - P r - ?)•

QiE. D.

2 3°

BOOK VI. PROP. XII. PROB.

and
take
and
alfo

draw

and

O find a fourth pro¬
portional to three
given lines

«iMMI a ■ I4M

Draw

9

9

9

- II

(B. i. pr. 31);
is the fourth proportional.

.... I .

making any angle;

On account of the parallels.

(B. 6. pr. 2);

f 4N«tB»BSeiR»* 1 f

but < mmmmmmmmmmmrn. V SS <

l J

| (conft.);

■ ■IliRflMva

• •
• •

(B. 5. pr. 7).

Q^E. D.

BOOK VI. PROP. XIII. PROB.

2 3 l

O find, a mean propor¬
tional between two given
firaight lines

* e «« mm «w *s m «■*■•*»■» ■» 1

Draw any ftraight line »»
make 1

and zz * bifedt — -. ...„ •

and from the point of bife&ion as a centre, and half the

line as a radius, defcribe a femicircle
draw ——— X - 1 '"Ui-...-
—— is the mean proportional required.

Draw and

is a right angle (B. 3. pr. 31),
is X from it upon the oppoiite iide,
is a mean proportional between

- and — (B. 6. pr. 8),

and .% between . an d . (conft.).

Q- E. D

BOOK VI. PROP. XIV. PHEOR.

232

I.

parallelograms

1 which have one angle in each equal ,
have the Jides about the equal angles
reciprocally proportional

II.

And parallelograms which have one angle in each equal ,
and the Jides about them reciprocally proportional , are equal.

Let ■ and — ; and

and mmm—mmmm be fo placed that ' 1 ——
and —- may be continued right lines. It is evi¬

dent that they may aflume this pofition. (B. i. prs. 13, 14,

Complete

• •

BOOK VL PROP. XIV. THEOR.

2 33

(B. 6. pr. i.)

The fame conftrudtion remaining :

and

r

%

(B. 6. pr. i.)

(hyp-)

(B. 6. pr. i.)

V %

(B. 5. pr. 11.)

(B. 5. pr. 9).

Q^E. D.

H H

234

BOOK VI. PROP. XV. THEOR.

I.

QUAL triangles , which have
one angle in each equal

4

, have the

/ides about the equal angles reciprocally
proportional

II.

And two triangles which have an angle of the one equal to
an angle of the other , and the fides about the equal angles reci¬
procally proportional , are equal.

I.

Let the triangles be fo placed that the equal angles

may be vertically oppofite, that is to fay,

fo that ii ■ .....pi.1i.. and — may be in the fame

flraight line. Whence alfo - .. 111 ■ and mull

be in the fame Ilraight line. (B. 1. pr. 14.)

(B. 6. pr. 1.)

(B. 5. pr. 7.)

• *

(B. 6. pr. 1 .)

BOOK VI. PROP. XV. THEOR.

2 35

(B. 5. pr. 11.)

II.

Let the fame conftruction remain, and

(B. 6. pr. 1.)

(B. 6. pr. 1.)

(B. 5- pr. n);

(B. 5. pr. 9.)

Q. E. D.

236 BOOK VI. PROP. XVI, THEOR.

8BFB1V site ear Bis on m mmnm

PART I.

¥ four Jiraight lines be proportional

( - : -

Mi • *

9 9

I at ivir

■ EUBBBKniiBB j ?

the re Bangle ( - X ■■■■■■*■■■) contained

by the extremes , is equal to the reBangle
— x . ) contained by the means.

PART II.

And if the reB¬
angle contained by
the extremes be equal
to the reBangle con¬
tained by the means ,
the fourftraight lines
are proportional.

PART I.

From the extremities of and draw

and ■ ■ ■■■■■ ■■— _L to them and «
and *«••**«»•■— refpedtively: complete the parallelograms

and

( h yp-)

(con ft.)

BOOK VI. PROP. XVI. THEOR .

that is, the re&angle contained by the extremes, equal to
the rectangle contained by the means.

PART II.

Let the fame conftrudtion remain ; becaufe

ISSKAIRBBBB

OKSBCKSSS^

Q. E. D.

Oo

238 BOOK VI. PROP. XVII. THEOR .

PART I

F three Jlraight lines be pro¬
portional ( — M l ® - .-

reB angle under the extremes
is equal to the fquare of the mean.

PART II.

And if the re Bangle under the ex¬
tremes be equal to the fquare of the mean ,
the three fraight lines are proportional.

PART I.

Afliime

fince

then

*

• •

x

• *
• •

• •

, and

x

9

9

(B. 6. pr. 16).

But

x

’ 9

X

or

: - 2 ; therefore, if the three ftraight lines are

proportional, the re&angle contained by the extremes is
equal to the fquare of the mean.

PART II.

Aflume

- x ■

, 9 then

x —

• •

• ©
• •

(B. 6. pr. 16), and

Q. E. D.

BOOK VI. PROP. XVIII. THEOR.

N a given Jiraight line (■

■)

to confiruB a reBilinear figure
fimilar to a given one (

and fimilarly placed.

Refolve the given figure into triangles by

drawing the lines ------- and

At the extremities of —■ make

^ = -1^. and ^ ;

again at the extremities of

make zz

in like manner make

1 ^ = ^\ :

V - ^ and ^ = V

Then

is limilar to

It is evident from the conftrudtion and (B. i. pr. 32) that
the figures ate equiangular j and fince the triangles

w and

are equiangular; then by (B. 6. pr. 4),

• @

• «

1 «>■!«■»■ mm*

and

• •

240 BOOK VI. PROP . XVIII. THEOR.

Again, becaufe

and

5

are equiangular.

• mm»n-amatmmmjm ® ® Hi mi

® # •

ilil

ex aequali.

© <© ©

(B. 6. pr. 22.)

In like manner it may be thown that the remaining fides
of the two figures are proportional.

by (B. 6. def. i.)
is fimilar to

and fimilarly fituated; and on the given line

E. D.

BOOK VL PROP. XIX. THEOR.

241

IMILAR trian -

j”

(

another in the duplicate ratio
of their homologous fides .

Let

and •

and

be equal angles, and

homologous iides of the fimilar triangles

and | an d on »■»■ ■■ the greater

of thefe lines take ■««.. a third proportional, fo that

illlll

• •
• •

draw

\

(B. 6. pr. 4);

• •

(B. 5. pr. 16, alt.),

but

(conft.).

• _

11

*

• •

■■■■■■

• •

confe-

242

BOOK VL PROP. XIX. THEOR.

quently

for they have the fides about

and -jik

the equal angles and reciprocally proportional

(B. 6. pr. 15);

but

A

(B. 5 pr. 7);

(B. 6. pr. 1),

mmm Hmmmm

A

■■■■1

that is to fay, the triangles are to one another in the dupli¬
cate ratio of their homologous fides
and ■■■*■■■*■■■■■■■* (B. def. 11^)«

Q. E. D.

BOOK VI. PROP. XX. THEOR.

2 43

IMILAR poly¬
gons may be di¬
vided into the
Y fame number of
fimilar triangles , each\fmilar
fair of which are propor¬
tional to the polygons; and
the polygons are to each other
in the duplicate ratio of their
homologous fides.

Draw ■■ ■ ■■ and

and -

7

and - ? refolving

the polygons into triangles.
Then becaufe the polygons

are fimilar,
and ..

and

4 = 4

b “' w = ♦

are fimilar, and =

(B. 6. pr. 6);

becaufe they are angles of fimilar poly¬

gons ; therefore the remainders

and

hence ««

are equal;

9

on account of the fimilar triangles,

2 44

BOOK VI. PROP . XX THEOR.

and

on account of the limilar polygons,

■WIIIIVBIBB

ex aequali (B. 5. pr. 22), and as thefe proportional lides
contain equal angles, the triangles ^ and

>

are limilar (B. 6. pr. 6).

In like manner it may be Ihown that the

triangles

But

▼

is to
to --<

and

are limilar.

in the duplicate ratio of
(B. 6. pr. 19), and

is to

>

in like manner, in the duplicate
ratio of «■■■■■•■■■■ to ;

A

(B. 5. pr. 11);

Again

is to

in the duplicate ratio of

to

, and

is to

W

in

BOOK PL PROP. XX. THEOR.

2 45

the duplicate ratio of ■ ■ ■ ■.. to —

and as one of the antecedents is to one of the confequents,
fo is the fum of all the antecedents to the fum of all the
confequents ; that is to fay, the limilar triangles have to one
another the fame ratio as the polygons (B„ 5. pr. 12).

But

is to

in the duplicate ratio of

to

Q. E. D

246 BOOK VI. PROP. XXI. THEOR.

ECTILINEAR fig

ures

and

which arejimilar to the famefigure (
are fimilar alfo to each other .

Since and are fimi¬

lar, they are equiangular, and have the
lides about the equal angles proportional
(B. 6. def. 1); and fince the figures

and are alfo fimilar, they

are equiangular, and have the lides about the equal angles

proportional; therefore and I- ^ are alfo

equiangular, and have the lides about the equal angles pro¬
portional (B. 5. pr. 11), and are therefore fimilar.

Q. E. D.

BOOK VI. PROP. XXII. THEOR.

247

PART I.

F fourJiraight lines be pro¬
portional (■

: ), the

Jimilar redlilinear figures
fimilarly described on them are alfio pro¬
portional.

PART II.

And if four Jimilar rectilinear
figures , fimilarly defcribed on Jour
Jiraight lines , be proportional , the
Jiraight lines are aljo proportional.

PART I.

Take -. a third proportional to ——

and .. 9 and a third proportional

to . .— and (B. 6. pr. 11);

fince :: " , — ■ : (hyp.),

■■ 1 111 * .. :: « ■■■ : •»«■■■••*** (conft.)

ex squali.

© «
9 9

I

9

248 BOOK VI. PROP. XXII. THEOR.

(B. 5. pr. ji).

PART II.

Let the fame conftru&ion remain :.

(B. 5. pr. 11).

(conft.)

E. D,

BOOK VI. PROP. XXIII. THEOR.

249

QUIANGULAR parallel¬
ograms ( m and

) are to one another
in a rath compounded of the ratios of
their fides.

Let two of the fides - and
about the equal angles be placed
fo that they may form one straight
line.

Since +

and = ▼ (hyp.),

= m

and

Jfe +

— and

form one ftraight line

(B. 1. pr. 14) ;
complete j/j .

Since

Ml

and

#

(B. 6. pr. 1),

(B.6. pr. 1),

has to
to

a ratio compounded of the ratios of
and of — i n to - .

K K.

(fE. D.

250

BOOK VI. PROP. XXIV. THEOR.

N any parallelogram (,
the parallelograms ( B

B

and y / ) which are about
the diagonal are Jimilar to the whole , and
to each other.

As

a*

a

have

common angle they are equiangular;
but becaufe ■■ ■■■■—■ ||

i«n

and

are limilar (B. 6. pr. 4),

• «
• •

iltlllf

and the remaining oppotite tides are equal to thofe,

B

and b/ / have the tides about the equal
angles proportional, and are therefore timilar.

In the fame manner it can be demontirated that the

rB

B

parallelograms rl / and W / are timilar.
Since, therefore, each of the parallelograms

and

B

is timilar to F / / 9 they are timilar
to each other.

Q. E. D.

BOOK VI. PROP. XXV. PROB.

251

O defcribe a reSlilinear figure ,
which fhall be fimilar to a given

reftilinearfigure ( / ), and

equal to another

Upon defcribe

and upon - defcribe _

and having

A

(B. 1. pr. 45), and then

and ■■■■■■■■■■ will lie in the fame ftraight line
(B. 1. prs. 29, 14),

Between

and »»«hh» find a mean proportional
(B. 6. pr. 13), and upon

defcribe . fimilar to

and fimilarly fituated.

Then

(B. 6. pr. 20);

BOOK VI. PROP . XXV . PPOP.

252

but

©

© ©

• #
• 9

9 9
9 9

but

and # \

□ = 9

= □

mmmmmmmmm (B. 6 . $T. l)j

: |_| (B-5.pr.11);

SI (conft.),

(B. 5. pr. 14);

and _I zz m (conft.); confequently,

which is iimilar to is alfo zz ti^,

Q. E. D.

BOOK VI. PROP. XXVI. THEOR.

2 53

F Jimilar and Jimilarly
pojited parallelograms

have a common angle , they are about
the fame diagonal.

For, if poffible, let

be the diagonal of

draw 11

(B. i. pr. 31).

Since

O

and

are about the fame

A

diagonal , and have common,

they are fimilar (B. 6. pr. 24) ;

• •

but

ttiattr.v • •
• »

( h yp-)>

>■. *

9

ta 1 »*•

and

tmm m-mm

(B. 5. pr. 9.),

which is abfurd.

is not the diagonal of

0

in the fame manner it can be demonftrated that no other

line is except

Q. E. D.

BOOK VI. PROP. XXVII. THEOR.

254

F all the re Bangles
contained by the
fegments of a given
Jlraight line, the
greatejl is the fquare which is
defcribed on half the line.

Let

be the

unequal fegments,
equal fegments;

For it has been demonftrated already (B. 2. pr. 5), that
the fquare of half the line is equal to the redtangle con¬
tained by any unequal fegments together with the fquare
of the part intermediate between the middle point and the
point of unequal fection. The fquare defcribed on half the
line exceeds therefore the redtangle contained by any un¬
equal fegments of the line.

Q. E. D.

BOOK VI. PROP. XXVIII. PROP. 255

O divide a given
Jiraight line

fo that the rec¬
tangle contained by its segments
may be equal to a given area,
not exceeding the fquare of
half the line.

Let the given area be

Bifedt
make «•’
and if —

or

the problem is folved.

But if
mud

mil

then

( h yp-)-

Draw
make ■
with —

MUfll

or •«

as radius defcribe a circle cutting the

given line; draw

Then

x

+

(B. 2. pi. 5.) —

But

+

(B. 1. pr. 47);

BOOK VI. PROP. XXVIII. PROB.

. X -

mmammmm +

— - -- 2 ...

from both, take r—■—

and *»»«»

%

Biit —™>TT — .»«»»«•■• (conft.),

and ■■■■ ■— mrnmrn mmmmmm IS fo divided

R4

that

x

BOOK VI. PROP. XXIX. PROB. 257

Oproduce a given flraight

line ( .. ■ ■ ■■■■■■), fo

that the re Bangle con¬
tained by the fegments
between the extremities of the given
line and the point to which it is pro¬
duced, may be equal to a given area,
i. e. equal to the fquare on —>.

Make - - — ......... 9 and

draw _L --------- zz —— 2

draw 111 11 ; and

with the radius f defcribe a circle

meeting produced.

Then

But

and

1 1 2 (B. 2. pr. 6 .) HI ■■■■ m 1 mm 2 .

2 z: ——•««— 3 -J- .. 2 (B. 1. pr.47.)

■ m X • -j- ............ 2 zz

2 4. 2

I ' ™ 9

from both take ..........

but *»—

*• 2 =z the given area.

Q±E. D.

L L

258 BOOK VI. PROP. XXX. PROP.

O cut a given finite firaight line ( ■»■ ■■ »»» )
in extreme and mean ratio.

On

defcribe the fquare

(B. i. pr. 46); and produce

. x ».

fo that

(B. 6. pr. 29);

take

and draw —

meeting

II

II

(B. 1. pr. 31).

Then

■ =

X

and is

and if from both thefe equals

be taken the common part 9
□ 9 which is the fquare of ———— ?
will be = |; | , which is = ——— X

and

is divided in extreme and mean ratio.
(B. 6 . def. 3).

Q. E. D.

BOOK VI. PROP. XXXI. THEOR. 259

F any Jimilar rectilinear
figures be fimilarly defcribed
on the fides of a right an¬
gled triangle ( ), the figure

defcribed on the fide (» .. ) fiub-

tending the right angle is equal to the
fum of the figures on the other fides.

From the right angle draw

tO ■■■■«■»!

perpendicular

then

• •
• •

(B. 6. pr. 8).

(B. 6. pr. 20).

■ - ' I • c _______ ®

H ## ■flflfiltflnHHM # ttn ■ n aet

(B. 6. pr. 20).

Q. E. D.

260 BOOK VI. PROP. XXXII. THEOR

F two triangles (

), have two Jides pro¬

portional (

and be foplaced
at an angle that the homologous Jides are pa¬
rallel the remaining Jides ( and ) form

one right line.

Since 1 ■

n

CRIHRIRMIRR

:zz (B. i. pr. 29) ;

and alfo lince .. fj

WiMI M *•« *

= /^> (B. 1. pr. 29);

; and iince

• e

• •

RMM# «r

(hyp.) ;

the triangles are equiangular (B. 6. pr. 6);

but

£ + + A = J + + —

/T\ (B. i. pr. 32), and and

lie in the fame flraight line (B. i. pr. 14).

E. D.

BOOK VI. PROP. XXXIII. THEOR. 261

N equal circles (

OO

angles ,

whether at the centre or circumference ,
z/z the fame ratio to one another as the arcs

on which they ftand (
fo alfo are fedlors .

L-4-.

);

Take in the circumference of

O

of arcs 9 «—» ? &c. each

the circumference of

O

arcs ***•■ «** 9 ? &c. each

any number
zr ***■» , and alfo in
take any number of
=z 9 draw the

radii to the extremities of the equal arcs.

Then lince the arcs

the angles

/, 4\

&c. are all equal,

9 W 9 & c - are alfo equal (B. 3. pr. 27);

4

is the fame multiple of $ which the arc

is of

! and in the fame manner

4 .

is the fame multiple of ^ , which the arc
is of the arc

262 BOOK VI. PROP . XXXIII. THEOR.

Then it is evident (B. 3. pr. 27),

(or if m times

(or n times

then ^ (or m times <«=■**) zz, 33

. . (or n times );

9 (B. 5. def. 5), or the
angles at the centre are as the arcs on which they hand;
but the angles at the circumference being halves of the
angles at the centre (B. 3. pr. 20) are in the fame ratio
(B. 5. pr. 15), and therefore are as the arcs on which they
hand.

It is evident, that fedtors in equal circles, and on equal
arcs are equal (B. 1. pr. 4; B. 3. prs. 24, 27, and def. 9).
Hence, if the fedtors be fubhituted for the angles in the
above demonhration, the fecond part of the propohtion will
be ehablifhed, that is, in equal circles the fedtors have the
fame ratio to one another as the arcs on which they hand.

Q. E. D.

BOOK VI. PROP. A. THEOR.

263

F the right line
bifedling an external

Jide {wmm—mmm} produced, that whole producedfide (
and its external fegment (-«■»«■■■■) will be proportional to the
(ides ( and ), which contain the angle

adjacent to the external bifedled angle .

For if 1 * 1 ■■■ — ■1 be drawn || •*■■■>■«■« 9
then V = £> , (B. 1. pr. 29);

= > ( h yp-)>

= , (B. 1. pr. 29);

and “ 9 (B. 1. pr. 6),

and — ■■ j j® v■ ■■ ■ & j KBiftftviaMR

(B. 5. pr. 7) ;

But alfo,

■ ■ ■ ■ « saaH<Bvaa "' JJ « i>miwmia»»ia J

(B. 6. pr. 2) ;
and therefore

(B. 5. pr. n).

\

Q. E. D.

264

BOOK VI. PROP. B. THEOR.

F an angle of a triangle be bi-
febled by a flraight line, which
likewife cuts the bafe; the rec¬
tangle contained by the fides of
the triangle is equal to the re 51 angle con¬
tained by the fegments of the bafe, together
with the fquare of the ftraight line which
bifeSls the angle.

Let

be drawn, making

then fhall

■ ■ i«i «m

2

About

defcribe

(B. 4. pr. 5),

produce

to meet the circle, and draw

Since

and

4 = A

= ►

(hyp.).

(B. 3. pr. 21),

are equiangular (B. 1. pr. 32);

• »

ll

(B. 6. pr. 4);

BOOK VI. PROP. B. THEOR.

265

—.. X . . ' ■ 1 X

(B. 6. pr. 16.)

= x 1 ■ + M 2

(B. 2. pr. 3);

but .- X 1 = - X

( B - 3 - P r - 35 );

Q. E. D.

M M

266

BOOK VI. PROP . C. THEOR.

F from any angle of a triangle a
flraight line be drawn perpendi¬
cular to the bafe; the rectangle
contained by the fides of the tri¬
angle is equal to the re 61 angle contained by
the perpendicular and the diameter of the
circle defcribed about the triangle.

y\

fhall

From of

draW mnmmmmm, i ■ ■■ • thdl

X = X the

diameter of the defcribed circle.

Defcribe

O

(B. 4. pr. 5), draw its diameter

) and draw

_

and

5 then becaufe
(conft. and B. 3. pr. 31);

/ = (B. 3- pr. 21);

and

(B. 6. pr. 4);

9

(B. 6. pr. 16).

Q. E. D.

BOOK VI. PROP. D. THEOR.

267

HE re Bangle contained by the
diagonals of a quadrilateralfigure
infcribed in a circle , is equal to
both the reBangles contained by

its oppojite Jides.

be any quadrilateral

figure infcribed in

and draw

and

5 then

■■■■■■■■■

Make A = V (B. i.pr. 23),

^ ^ ; ,»d = (

(B. 3. pr. 21);

(B. 6. pr. 4);

and

x

X

becaufe

(B. 6. pr. 16); again,

A =

(conft.),

268

BOOK VI. PROP. D. THEOR .

and

(B. 3. pr. 21);

aa«aiMiBii

• »it»**»<*»*»« • 9

% • 9

(B. 6. pr. 4) ;

and • «-

x

■a iHBiiaiDii

X

(B. 6. pr. 16);
hut, from above.

x

x

X

X

+

»«wwmmmmnm *

X •mmmmmmm

(B. 2. pr. 1.

E. D.

THE END.

CHISWICK : PRINTED BY C. W1IITTINGHAM.

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